Document:

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                                                                   EXHIBIT 10.14

                               WARRANT CERTIFICATE

Number of Warrants: 250,000                                    Warrant No.   A-5

          This warrant certificate ("Warrant Certificate") certifies that, for
value received,

                         ECT MERCHANT INVESTMENTS CORP.

is the registered holder of the number of warrants (the "Warrants") set forth
above. Each Warrant entitles the holder thereof, at any time or from time to
time during the Exercise Period, to purchase from the Company one fully paid and
nonassessable share of Common Stock at the Exercise Price, subject to adjustment
as provided herein. The Warrants constitute, as of February 17, 2000, 1.0338% of
the outstanding Common Stock on a fully diluted basis including, for purposes of
such calculation, the Acquired Shares and the Warrant Shares. Initially
capitalized terms used but not defined herein shall have the meanings ascribed
to them in the Securities Purchase Agreement. This Warrant Certificate amends
and restates Warrant No. A-3 issued to ECT Merchant Investments Corp. as of
March 26, 1999, which in turn had amended and restated Warrant No. A-1 issued to
Enron Capital & Trade Resources Corp. on August 20, 1998.

          "Acquired Shares" means the shares of Common Stock acquired by Enron
North America Corp. and Joint Energy Development Investments II Limited
Partnership pursuant to Section 2.01 of the Securities Purchase Agreement.

          "Appraiser" has the meaning set forth in Section 13.

          "Appreciation Amount" has the meaning set forth in Section 1(b).

          "Common Stock" means the common stock, $.01 par value per share, of
the Company and such other class of securities as shall then represent the
common equity of the Company.

          "Company" means Brigham Exploration Company, a Delaware corporation.

          "Exercise Period" means the period of time between the Funding Date,
as defined in the Securities Purchase Agreement and 5:00 p.m. (New York City
time) on the Expiration Date.

          "Exercise Price" subject in all circumstances to adjustment in
accordance with Section 2, means $______, which represents the average volume
weighted Price during the twenty (20) calendar day period that begins February
22, 2000 and ends March 12, 2000.

          "Expiration Date" means August 22, 2008.

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          "Dissenting Notice" has the meaning set forth in Section 13.

          "Fair Market Value" has the meaning set forth in Section 13.

          "Funding Date" is defined in the Securities Purchase Agreement.

          "IPO" shall mean the initial public offering of securities of the
Company consummated on May 24, 1997, pursuant to a registration statement filed
under the Securities Act of 1933, as amended, and the rules and regulations
promulgated thereunder.

          "Issuance Date" means August 20, 1998.

          "Notes" means the Senior Subordinated Secured Notes issued pursuant to
the Securities Purchase Agreement.

          "Person" means any individual, corporation, company, partnership,
joint venture, trust, limited liability company, unincorporated organization or
government or any agency, instrumentality or political subdivision thereof, or
any other form of entity.

          "Price" means the average of the "high" and "low" prices as reported
in The Wall Street Journal's listing for such day (corrected for obvious
typographical errors) or if such shares are not reported in such listing, the
average of the reported "high" and "low" sales prices on the largest national
securities exchange (based on the aggregate dollar value of securities listed)
on which such shares are listed or traded, or if such shares are not listed or
traded on any national securities exchange, then the average of the reported
"high" and "low" sales prices for such shares in the over-the-counter market, as
reported on the National Association of Securities Dealers Automated Quotations
System, or, if such prices shall not be reported thereon, the average of the
closing bid and asked prices so reported, or, if such prices shall not be
reported, then the average of the closing bid and asked prices reported by the
National Quotations Bureau Incorporated, or, in all other cases, the Fair Market
Value as determined in accordance with Section 13 below. The "average" Price per
share for any period shall be determined by dividing the sum of the Prices
determined for the individual trading days in such period by the number of
trading days in such period.

          "Publicly Traded" means, with respect to the Common Stock, that such
securities are listed for trading on the New York Stock Exchange, Inc., the
American Stock Exchange, Inc. or the NASDAQ National Market. The Common Stock
shall also be deemed to be Publicly Traded if the Common Stock is included in
the NASDAQ SmallCap Market.

          "Securities Purchase Agreement" means the Securities Purchase
Agreement, dated as of August 20, 1998, among the Company, Enron North America
Corp. (formerly named Enron Capital

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& Trade Resources Corp.), and Joint Energy Development Investments II Limited
Partnership, individually and as agent, as amended from time to time.

          "Warrant Shares" means the shares of Common Stock and other securities
receivable upon exercise of the Warrants.

          "Warrants" means the Warrants issued by the Company to Enron North
America Corp. and Joint Energy Development Investments II Limited Partnership
pursuant to the Securities Purchase Agreement and any Warrants issued upon the
transfer thereof or in substitution therefor.

          1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in whole or
in part, at any time or from time to time, during the Exercise Period, by
presentation and surrender to the Company at its address set forth in Section 9
of (i) this Warrant Certificate with the Election To Exercise, attached hereto
as Exhibit A-1, duly completed and executed, and (ii) payment of the Exercise
Price for the number of Warrants being exercised. At the option of the holder
hereof, payment of the Exercise Price shall be made (w) by wire transfer of
funds, (x) by check payable to the order of the Company, (y) by application of
any Warrant Shares or credit against any Notes, as provided below, or (z) by any
combination of such methods.

If the holder of this Warrant Certificate at any time exercises less than all
the Warrants, the Company shall issue to such holder a warrant certificate
identical in form to this Warrant Certificate, but evidencing a number of
Warrants equal to the number of Warrants originally represented by this Warrant
Certificate less the number of Warrants previously exercised. Likewise, upon the
presentation and surrender of this Warrant Certificate to the Company at its
address set forth in Section 9 and at the request of the holder, the Company
will, without expense, at the option of the holder, issue to the holder in
substitution for this Warrant Certificate one or more warrant certificates in
identical form and for an aggregate number of Warrants equal to the number of
Warrants evidenced by this Warrant Certificate.

               (b) In lieu of exercise in accordance with paragraph (a) above,
the Warrants represented hereby may be exercised, in whole or in part, by
presentation and surrender at the office of the Company specified herein of this
Warrant Certificate with the "Alternative Election to Exercise", attached hereto
as Exhibit A-2, duly completed and executed. Upon such exercise, the holder
shall be entitled to receive from the Company, for each share of Common Stock
issuable upon exercise of each Warrant being exercised, shares of Common Stock
with a value equal to the amount by which either the Price or, in the event that
the Common Stock is not Publicly Traded, the Fair Market Value of one share of
Common Stock on the date of exercise exceeds the Exercise Price (the
"Appreciation Amount"). If the Common Stock is Publicly Traded on the date of
exercise of Warrants pursuant to this Section 1(b), the Company shall deliver
(or cause its transfer agent to deliver) certificates representing shares of
Common Stock issuable upon such exercise within five business days following the
date of exercise. If the Common Stock is not Publicly Traded on the date

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of exercise of Warrants pursuant to this Section 1(b), such election may be
withdrawn by the holder at any time prior to the third business day following
the date on which the holder receives notice of the final determination of Fair
Market Value pursuant to Section 15. Unless the holder timely withdraws its
exercise, the Company shall deliver (or cause its transfer agent to deliver)
certificates representing Common Stock so issuable within 10 business days
following the date of determination of the Fair Market Value. In lieu of any
fractional shares otherwise issuable upon exercise pursuant hereto, the Company
shall pay to the holder cash in an amount equal to the Price, or in the event
the Company is not Publicly Traded, the Fair Market Value of such fractional
share.

               (c) Upon the exercise of this Warrant in whole or in part by the
holder of any Notes, such holder may, at its option, deliver written notice to
the Company that the holder has elected to apply some or all of the Exercise
Price required upon such exercise against all or any specified principal amount
of and/or accrued but unpaid interest on, such Notes, in which case the Company
will accept the application of the Exercise Price against such specified
principal and/or interest amount in lieu of a like amount of cash payment.

               (d) To the extent that the Warrants have not been exercised at or
prior to the Expiration Date, such Warrants shall expire and the rights of the
holder shall become void and of no effect.

          2. ANTIDILUTION ADJUSTMENTS. The shares of Common Stock purchasable on
exercise of the Warrants are shares of Common Stock as constituted as of the
Issuance Date. The number and kind of securities purchasable upon the exercise
of the Warrants, and the Exercise Price, shall be subject to adjustment from
time to time upon the happening of certain events, as follows:

               (a) Mergers, Consolidations and Reclassifications. In case of any
reclassification or change of outstanding securities issuable upon exercise of
the Warrants at any time after the Issuance Date (other than a change in par
value, or from par value to no par value, or from no par value to par value or
as a result of a subdivision or combination to which subsection 2(b) applies),
or in case of any consolidation or merger of the Company with or into another
entity or other person (other than a merger with another entity or other person
in which the Company is the surviving corporation and which does not result in
any reclassification or change in the securities issuable upon exercise of this
Warrant Certificate), the holder of the Warrants shall have, and the Company, or
such successor corporation or other entity, shall covenant in the constituent
documents effecting any of the foregoing transactions that such holder does
have, the right to obtain upon the exercise of the Warrants, in lieu of each
share of Common Stock, other securities, money or other property theretofore
issuable upon exercise of a Warrant, the kind and amount of shares of stock,
other securities, money or other property receivable upon such reclassification,
change, consolidation or merger by a holder of the shares of Common Stock, other
securities, money or other property issuable upon exercise of a Warrant if the
Warrants had been exercised immediately prior to such reclassification, change,
consolidation or merger. The constituent documents effecting any such

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reclassification, change, consolidation or merger shall provide for adjustments
which shall be as nearly equivalent as may be practicable to the adjustments
provided in this subsection 2(a). The provisions of this subsection 2(a) shall
similarly apply to successive reclassifications, changes, consolidations or
mergers.

               (b) Subdivisions and Combinations. If the Company, at any time
after the Issuance Date, shall subdivide its shares of Common Stock into a
greater number of shares, the Exercise Price in effect immediately prior to such
subdivision shall be proportionately reduced, and the number of shares of Common
Stock purchasable upon exercise of the Warrants shall be proportionately
increased, as at the effective date of such subdivision, or if the Company shall
take a record of holders of its Common Stock for such purpose, as at such record
date, whichever is earlier. If the Company, at any time after the Issuance Date,
shall combine its shares of Common Stock into a smaller number of shares, the
Exercise Price in effect immediately prior to such combination shall be
proportionately increased, and the number of shares of Common Stock purchasable
upon exercise of the Warrants shall be proportionately reduced, as at the
effective date of such combination, or if the Company shall take a record of
holders of its Common Stock for purposes of such combination, as at such record
date, whichever is earlier.

               (c) Dividends and Distributions. If the Company at any time after
the Issuance Date shall declare a dividend on its Common Stock payable in stock
or other securities of the Company to the holders of its Common Stock, the
holder of this Warrant Certificate shall, without additional cost, be entitled
to receive upon any exercise of a Warrant, in addition to the Common Stock to
which such holder would otherwise be entitled upon such exercise, the number of
shares of stock or other securities which such holder would have been entitled
to receive if he had been a holder immediately prior to the record date for such
dividend (or, if no record date shall have been established, the payment date
for such dividend) of the number of shares of Common Stock purchasable on
exercise of such Warrant immediately prior to such record date or payment date,
as the case may be.

               (d) Certain Issuances of Securities. If the Company at any time
after the Issuance Date shall issue any additional shares of Common Stock
(otherwise than as provided in paragraphs (a) through (c) of this Section 2) at
a price per share less than the average Price per share of Common Stock for the
20 trading days immediately preceding the date of the authorization of such
issuance (the "Market Price") by the Board of Directors, then the Exercise Price
upon each such issuance shall be adjusted to that price determined by
multiplying the Exercise Price by a fraction:

                    i. the numerator of which shall be the sum of (1) the number
          of shares of Common Stock outstanding immediately prior to the
          issuance of such additional shares of Common Stock multiplied by the
          Market Price, and (2) the consideration, if any, received by the
          Company upon the issuance of such additional shares of Common Stock,
          and

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                    ii. the denominator of which shall be the Market Price
          multiplied by the total number of shares of Common Stock outstanding
          immediately after the issuance of such additional shares of Common
          Stock.

          No adjustments of the Exercise Price shall be made under this
paragraph (d) upon the issuance of any additional shares of Common Stock that
(y) are issued pursuant to thrift plans, stock purchase plans, stock bonus
plans, stock option plans, employee stock ownership plans and other incentive or
profit sharing arrangements for the benefit of employees ("Employee Benefit
Plans") that otherwise would cause an adjustment under this paragraph (d);
provided that the aggregate number of shares of Common Stock so issued
(including the shares issued pursuant to any options, rights or warrants or
convertible or exchangeable securities issued under such Employee Benefit Plans
containing the right to purchase shares of Common Stock) pursuant to Employee
Benefit Plans after the closing date of the IPO, as adjusted for any stock
splits, stock dividends or subdivisions or combinations of Common Stock prior to
the Expiration Date, shall not in the aggregate exceed 5% of the Company's
outstanding Common Stock at the time of such issuance; or (z) are issued
pursuant to any Common Stock Equivalent (as hereinafter defined) (i) if upon the
issuance of any such Common Stock Equivalent, any such adjustments shall
previously have been made pursuant to paragraph (e) of this Section 2 or (ii) if
no adjustment was required pursuant to paragraph (e) of this Section 2.

               (e) Common Stock Equivalents. If the Company shall, after the
Issuance Date, issue any security or evidence of indebtedness which is
convertible into or exchangeable for Common Stock ("Convertible Security"), or
any warrant, option or other right to subscribe for or purchase Common Stock or
any Convertible Security, other than pursuant to Employee Benefit Plans
(together with Convertible Securities, "Common Stock Equivalent"), or if, after
any such issuance, the price per share for which additional shares of Common
Stock may be issuable thereunder is amended, then the Exercise Price upon each
such issuance or amendment shall be adjusted as provided in subsection (d) on
the basis that (i) the maximum number of additional shares of Common Stock
issuable pursuant to all such Common Stock Equivalents shall be deemed to have
been issued as of the earlier of (a) the date on which the Company shall enter
into a firm contract for the issuance of such Common Stock Equivalent, or (b)
the date of actual issuance of such Common Stock Equivalent; and (ii) the
aggregate consideration for such maximum number of additional shares of Common
Stock shall be deemed to be the minimum consideration received and receivable by
the Company for the issuance of such additional shares of Common Stock pursuant
to such Common Stock Equivalent; provided, however, that no adjustment shall be
made pursuant to this subsection (e) unless the consideration received and
receivable by the Company per share of Common Stock for the issuance of such
additional shares of Common Stock pursuant to such Common Stock Equivalent is
less than the Market Price. No adjustment of the Exercise Price shall be made
under this subsection (e) upon the issuance of any Convertible Security which is
issued pursuant to the exercise of any warrants or other subscription or
purchase rights therefor, if any adjustment shall previously have been made in
the Exercise Price then in effect upon the issuance of such warrants or other
rights pursuant to this subsection (e).

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               (f) Miscellaneous. The following provisions shall be applicable
to the making of adjustments in the Exercise Price hereinbefore provided in this
Section 2:

                    i. The consideration received by the Company shall be deemed
          to be the following: (I) to the extent that any additional shares of
          Common Stock or any Common Stock Equivalent shall be issued for cash
          consideration, the consideration received by the Company therefor, or,
          if such additional shares of Common Stock or Common Stock Equivalent
          are offered by the Company for subscription, the subscription price,
          or, if such additional shares of Common Stock or Common Stock
          Equivalent are sold to underwriters or dealers for public offering
          without a subscription offering, the initial public offering price, in
          any such case excluding any amounts paid or receivable for accrued
          interest or accrued dividends and without deduction of any
          compensation, discounts, commissions or expenses paid or incurred by
          the Company for and in the underwriting of, or otherwise in connection
          with, the issue thereof; (II) to the extent that such issuance shall
          be for a consideration other than cash, then, except as herein
          otherwise expressly provided, the fair value of such consideration at
          the time of such issuance as determined in good faith by the Board of
          Directors, as evidenced by a certified resolution of the Board of
          Directors delivered to the holder of this Warrant Certificate setting
          forth such determination. The consideration for any additional shares
          of Common Stock issuable pursuant to any Common Stock Equivalent shall
          be the consideration received by the Company for issuing such Common
          Stock Equivalent, plus the additional consideration payable to the
          Company upon the exercise, conversion or exchange of such Common Stock
          Equivalent. In case of the issuance at any time of any additional
          shares of Common Stock or Common Stock Equivalent in payment or
          satisfaction of any dividend upon any class of stock other than Common
          Stock, the Company shall be deemed to have received for such
          additional shares of Common Stock or Common Stock Equivalent (which
          shall not be deemed to be a dividend payable in, or other distribution
          of, Common Stock under subsection (c) above) consideration equal to
          the amount of such dividend so paid or satisfied.

                    ii. Upon the expiration of the right to convert, exchange or
          exercise any Common Stock Equivalent the issuance of which effected an
          adjustment in the Exercise Price, if any such Common Stock Equivalent
          shall not have been converted, exercised or exchanged, the number of
          shares of Common Stock deemed to be issued and outstanding because
          they were issuable upon conversion, exchange or exercise of any such
          Common Stock Equivalent shall no longer be computed as set forth
          above, and the Exercise Price shall forthwith be readjusted and
          thereafter be the price which it would have been (but reflecting any
          other adjustments in the Exercise Price made pursuant to the
          provisions of subsection (d) after the issuance of such Common Stock
          Equivalent) had the adjustment of the Exercise Price made upon the
          issuance or sale of such Common Stock Equivalent been made on the
          basis of the issuance only of the number of additional shares of
          Common Stock actually issued upon exercise, conversion or exchange of
          such Common Stock Equivalent and thereupon only the

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          number of additional shares of Common Stock actually so issued shall
          be deemed to have been issued and only the consideration actually
          received by the Company (computed as in subparagraph (i) of this
          paragraph (f)) shall be deemed to have been received by the Company.

                    iii. The number of shares of Common Stock at any time
          outstanding shall not include any shares thereof then directly or
          indirectly owned or held by or for the account of the Company or its
          wholly owned subsidiaries.

                    iv. For the purposes of this Section 2, the term "shares of
          Common Stock" shall mean shares of (i) the class of stock designated
          as the Common Stock at the date hereof or (ii) any other class of
          stock resulting from successive changes or reclassifications of such
          shares consisting solely of changes in par value, or from par value to
          no par value, or from no par value to par value. If at any time,
          because of an adjustment pursuant to subsection (a), the Warrants
          shall entitle the holders to purchase any securities other than shares
          of Common Stock, thereafter the number of such other securities so
          purchasable upon exercise of each Warrant and the Exercise Price of
          such securities shall be subject to adjustment from time to time in a
          manner and on terms as nearly equivalent as practicable to the
          provisions with respect to the Common Stock contained in this Section
          2.

               (g) Calculation of Exercise Price. The Exercise Price in effect
from time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

          3. NOTICE OF ADJUSTMENTS. Whenever the Exercise Price or the number of
shares of Common Stock is required to be adjusted as provided in Section 2, the
Company shall forthwith compute the adjusted Exercise Price or the number of
shares of Common Stock issuable and shall prepare and mail to the holder hereof
a certificate setting forth such adjusted Exercise Price or such number of
shares of Common Stock, showing in reasonable detail the facts upon which the
adjustment is based.

          4. VOLUNTARY REDUCTION. (a) The Company may at its option, but shall
not be obligated to, at any time during the term of the Warrants, reduce the
then current Exercise Price by any amount selected by the Board of Directors;
provided that if the Company elects so to reduce the then current Exercise
Price, such reduction shall be irrevocable during its effective period and
remain in effect for a minimum of 30 days following the date of such election,
after which time the Company may, at its option, reinstate the Exercise Price in
effect prior to such reduction. Whenever the Exercise Price is reduced, the
Company shall mail to the holder a notice of the reduction at least 30 days
before the date the reduced Exercise Price takes effect, stating the reduced
Exercise Price and the period for which such reduced Exercise Price will be in
effect.

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               (b) The Company may make such decreases in the Exercise Price, in
addition to those required or allowed by this Section 4, as shall be determined
by it, as evidenced by a certified resolution of the Board of Directors
delivered to the holders, to be advisable to avoid or diminish any income tax to
the holder resulting from any dividend or distribution of stock or issuance of
rights or warrants to purchase or subscribe for stock or from any event treated
as such for income tax purposes.

          5. NOTICES TO WARRANT HOLDERS. In the event:

               (a) the Company shall authorize any consolidation or merger to
which the Company is a party and for which approval of any stockholders of the
Company is required, or of the conveyance or sale of all or substantially all of
the assets of the Company, or of any reclassification or change of the Common
Stock or other securities issuable upon exercise of the Warrants (other than a
change in par value, or from par value to no par value, or from no par value to
par value or as a result of a subdivision or combination), or a tender offer or
exchange offer for shares of Common Stock (or other securities issuable upon the
exercise of the Warrants); or

               (b) the Company shall declare any dividend (or any other
distribution) on the Common Stock or any other class of its capital stock; or

               (c) the Company shall authorize the granting to the holders of
Common Stock or any other class of its capital stock of rights or warrants to
subscribe for or purchase any shares of any class or series of capital stock or
any other securities convertible into or exchangeable for shares of stock; or

               (d) of the voluntary or involuntary dissolution, liquidation or
winding up of the Company;

          then the Company shall cause to be sent to the holder hereof, at least
30 days prior to the applicable record date hereinafter specified, or promptly
in the case of events for which there is no record date, a written notice
stating (x) the date for the determination of the holders of record of shares of
Common Stock (or other securities issuable upon the exercise of the Warrants)
entitled to receive any such dividends or other distribution, (y) the initial
expiration date set forth in any tender offer or exchange offer for shares of
Common Stock (or other securities issuable upon the exercise of the Warrants),
or (z) the date on which any of the events specified in subsections (a)-(d) is
expected to become effective or consummated, and the date as of which it is
expected that holders of record of shares of Common Stock (or other securities
issuable upon the exercise of the Warrants) shall be entitled to exchange such
shares for securities or other property, if any, deliverable upon any such
event. Failure to give such notice or any defect therein shall not affect the
legality or validity of any such event, or the vote upon any such action.

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          6. REPORTS TO WARRANT HOLDERS. The Company will cause to be delivered,
by first-class mail, postage prepaid, to the holder at such holder's address
appearing hereon, or such other address as the holder shall specify, a copy of
any reports delivered by the Company to the holders of Common Stock.

          7. COVENANTS OF THE COMPANY. The Company covenants and agrees that:

               (a) Until the Expiration Date, the Company shall at all times
reserve and keep available, free from preemptive rights, out of the aggregate of
its authorized but unissued Common Stock (and other securities), for the purpose
of enabling it to satisfy any obligation to issue shares of Common Stock (and
other securities) upon the exercise of the Warrants, the number of shares of
Common Stock (and other securities) issuable upon the exercise of such Warrants.

               (b) The Company shall pay all expenses, taxes and other charges
payable in connection with the preparation, issuance and delivery of new warrant
certificates on transfer of the Warrants.

               (c) All Common Stock (and other securities) which may be issued
upon exercise of the Warrants shall upon issuance be validly issued, fully paid,
non-assessable and free from all preemptive rights and all taxes, liens and
charges with respect to the issuance thereof, and will not be subject to any
restrictions on voting or transfer thereof except as set forth in any
stockholders agreement.

               (d) All original issue taxes payable in respect of the issuance
of shares of Common Stock to the registered holder hereof upon the exercise of
the Warrants shall be borne by the Company; provided, that the Company shall not
be required to pay any tax or charge imposed in connection with any transfer
involved in the issuance of any certificate representing shares of Common Stock
(and other securities) in any name other than that of the registered holder
hereof, and in such case the Company shall not be required to issue or deliver
any certificate representing shares of Common Stock (and other securities) until
such tax or other charge has been paid or it has been established to the
Company's satisfaction that no such tax or charge is due.

               (e) As soon as practicable after the receipt from the holder of
this Warrant Certificate of notice of the exercise of a number of warrants
sufficient to require a filing under the Hart-Scott-Rodino Antitrust
Improvements Act of 1976 and the rules, regulations and formal interpretations
thereunder, as amended from time to time (the "HSR Act"), but in any event no
later than the 10th business day after receipt of such notice, the Company will
(i) prepare and file with the Antitrust Division of the Department of Justice
(the "DOJ") and the Federal Trade Commission (the "FTC") the Notification and
Report Form (accompanied by all documentary attachments contemplated thereby)
required by the HSR Act, (ii) upon the request of the holder, request early
termination of the waiting period imposed by the HSR Act, (iii) coordinate and
cooperate with the

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holder in responding to formal and informal requests for additional information
and documentary material from the DOJ and the FTC in connection with such
filing, (iv) use its best efforts to take, or cause to be taken, all reasonable
action and to do, or cause to be done, all things necessary and appropriate to
permit the issuance to the holder of the shares of Common Stock issuable upon
the exercise of the warrants with respect to which any filing is required under
the HSR Act, and (v) reimburse the holder for the entire amount of any filing
fee or any other costs and expenses incurred by the holder in connection
therewith (including legal fees), or as required to be paid under the HSR Act.

               (f) QUOTATION ON NASDAQ. The Company shall maintain the
designation and quotations, or listing, of its Common Stock on the NASDAQ
national market (or on the New York Stock Exchange or the American Stock
Exchange) until the date on which none of the Warrants or Warrant Shares remain
outstanding.

          8. NO RIGHTS AS STOCKHOLDER. The holder of the Warrants shall not, by
virtue of holding such Warrants, be entitled to any rights of a stockholder of
the Company either at law or in equity, and the rights of the holder of the
Warrants are limited to those expressed herein.

          9. NOTICES. All notices provided for hereunder shall be in writing and
may be given by registered or certified mail, return receipt requested, telex,
telegram, telecopier, air courier guaranteeing overnight delivery of personal
delivery, if to the holder at the following address:

              ECT Merchant Investments Corp.
              1400 Smith Street
              Houston, Texas 77002
              Attention:  Donna Lowry
              Telecopier:  (713) 646-4039

          and, if to the Company:

              Brigham Exploration Company
              6300 Bride Point Parkway
              Building 2, Suite 500
              Austin, Texas 78730
              Attention: Curtis F. Harrell
              Telecopier: (512) 472-3400

          10. GOVERNING LAW. This Warrant Certificate shall be governed by and
construed in accordance with the laws of the State of Texas without regard to
principles of conflict of laws.

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          11. LOST, STOLEN, MUTILATED OR DESTROYED WARRANT CERTIFICATES. Upon
receipt by the Company of evidence reasonably satisfactory to it of the
ownership of and the loss, theft, destruction or mutilation of any Warrant
Certificate, then, in the absence of notice to the Company that such Warrant
Certificate has been acquired by a bona fide purchaser, the Company shall
execute and deliver, in exchange for or in lieu of the lost, stolen, destroyed
or mutilated Warrant Certificate, a substitute Warrant Certificate of the same
tenor and evidencing a like number of Warrants.

          12. ASSIGNMENT. The holder of this Warrant Certificate shall be
entitled, without obtaining the consent of the Company, to transfer or assign
its rights, title and interest in (and rights, title and interest under) this
Warrant Certificate in whole or in part to any Person or Persons. Upon surrender
of this Warrant Certificate to the Company, with the Transfer Form annexed
hereto as Exhibit B duly executed, the Company shall, without charge, execute
and deliver a new warrant certificate or warrant certificates, identical in form
to this Warrant Certificate, evidencing the number of Warrants being transferred
pursuant to the Transfer Form in the name of the assignee or assignees named in
such Transfer Form. If the holder's entire interest is not being assigned, the
Company shall, without charge, execute and deliver one or more new warrant
certificates identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number being transferred pursuant to the
Transfer Form, and this Warrant Certificate shall promptly be canceled. The
terms and provisions of this Warrant Certificate shall inure to the benefit of
the holder and its successors and assigns and shall be binding upon the Company
and its successors and assigns, including, without limitation, any Person
succeeding the Company by merger, consolidation or acquisition of all or
substantially all of the Company's assets.

          13. FAIR MARKET VALUE. Whenever the terms of this Warrant require a
determination of the fair market value of the Warrants or the Common Stock and,
at the time of such determination, the Common Stock or the Warrants, as the case
may be, are not Publicly Traded, such determination shall be made in good faith
by the Board of Directors of the Company within 30 days and the Company shall
notify the holder of such determination no later than two business days
following the date of such determination. If the holder disagrees with the fair
market value as so determined by the Board of Directors, such holder shall so
notify the Company within five business days after receipt of notice from the
Board of Directors of its determination and such holder shall include in such
notice (a "Dissenting Notice"), the holder's estimate of the fair market value.
In the event the holder and the Company are unable to reach agreement regarding
the fair market value with five business days after receipt by the Company of
the Dissenting Notice, then the fair market value shall be determined by an
independent third party, knowledgeable and experienced in the valuation of
private businesses, and who is mutually acceptable to the Company and the holder
(the "Appraiser"). In the event the Company and the holder are unable to agree
upon an Appraiser within two business days after receipt by the Company of the
Dissenting Notice, each party shall, within 2 business days, select an
investment banking firm and the two investment banking firms so selected shall,
within 5 business days, select a third investment banking firm to be the
Appraiser. The Appraiser shall make a

                                      -12-
<PAGE>   13

determination of fair market value within 20 business days of his appointment
and shall notify the Company and the holder of such determination, which shall
be binding on both parties. The costs associated with the determination of fair
market value shall be borne by the Company (as such value is determined by this
Section 15, the "Fair Market Value").

                                      -13-
<PAGE>   14

          IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to
be executed as of February 17, 2000, by the undersigned, thereunto duly
authorized.

                                       BRIGHAM EXPLORATION COMPANY

                                       By:
                                          --------------------------------------
                                             Curtis F. Harrell
                                             Chief Financial Officer

                                      -14-
<PAGE>   15

                                   EXHIBIT A-1

                              ELECTION TO EXERCISE
      [To be executed on exercise of the Warrants evidenced by this Warrant
                     Certificate pursuant to Section 1(a)]

TO:       Brigham Exploration Company

         The undersigned, the holder of the Warrants evidenced by the attached
Warrant Certificate, hereby irrevocably elects to exercise ______________ of
such Warrants, and herewith makes payment of ________________ ($___________)
representing the aggregate Exercise Price thereof, and requests that the
certificate representing the securities issuable hereunder be issued in the name
of _____________________ and delivered to __________________, whose address is
_____________________________.

The Exercise Price is being paid in the following manner: [ ] by bank draft or
cashier's check in the amount of ___________________ ($________), [ ] by
application of __________________ ($_________) of the outstanding principal
balance of the holder's Notes, and/or [ ] by application of _______________
($_______) of interest accrued on the holder's Notes.

         Dated:
               -----------

                           Name of Registered Holder:
                                                     ---------------------------
                           Signature:
                                     -------------------------------------------
                           Title:
                                 -----------------------------------------------

                           Address:
                                   ---------------------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -15-
<PAGE>   16

                                   EXHIBIT A-2

                        ALTERNATIVE ELECTION TO EXERCISE
      [To be executed on exercise of the Warrants evidenced by this Warrant
                     Certificate pursuant to Section 1(b)]

TO:       Brigham Exploration Company (the "Company")

          The undersigned holder of the Warrants evidenced by the attached
Warrant Certificate, hereby elects to exercise ____________ Warrants.

[ ]       The undersigned elects to receive shares of Common Stock of the
          Company in an amount equal to the Appreciation Amount (as defined in
          the Warrant), in accordance with Section 1(b) of the Warrant, and
          requests that the securities issuable hereunder be issued in the name
          of ___________, whose address is ____________________________________.

          Dated:
                ------------------

                                  Name of Registered Holder:
                                                            --------------------
                                  Signature:
                                            ------------------------------------
                                  Title:
                                        ----------------------------------------

                                  Address:
                                          --------------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -16-
<PAGE>   17

                                    EXHIBIT B

                                  TRANSFER FORM
  [To be executed only upon transfer of the Warrants evidenced by this Warrant
                                  Certificate]

          FOR VALUE RECEIVED, the undersigned hereby sells, assigns and
transfers unto ______________________________________________________
_________________ of the Warrants represented by the within Warrant Certificate,
together with all right, title and interest therein, and does hereby irrevocably
constitute and appoint _____________________________________ Attorney-in-Fact,
to transfer same on the books of the Company with full power of substitution in
the premises.

          Dated:
                ------------------

                                  Name of Registered Holder:
                                                            --------------------
                                  Signature:
                                            ------------------------------------
                                  Title:
                                        ----------------------------------------

                                  Address:
                                          --------------------------------------

WITNESS:

----------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -17-<PAGE>   1
                                                                   EXHIBIT 10.15

              SECURITIES PURCHASE AND REGISTRATION RIGHTS AGREEMENT

                                  by and among

                           Brigham Exploration Company

                                       and

                       GAP Coinvestment Partners II, L.P.,
                Special Situations Private Equity Fund, L.P. and
                            Aspect Resources, L.L.C.

                                February 22, 2000

<PAGE>   2

              SECURITIES PURCHASE AND REGISTRATION RIGHTS AGREEMENT

                                    RECITALS:

         (a) This SECURITIES PURCHASE AND REGISTRATION RIGHTS AGREEMENT (this
"Agreement") is dated as of February 22, 2000, by and among GAP Coinvestment
Partners II, L.P., Special Situations Private Equity Fund, L.P., and Aspect
Resources, L.L.C. (each individually, a "Buyer," and collectively, the "Buyers")
and Brigham Exploration Company, a Delaware corporation (the "Company"); and

         (b) The Company desires to issue to Buyers, and Buyers desire to
purchase, shares of the Company's common stock, par value $.01 per share and
warrants affording the right to purchase shares of the Company's common stock.

         NOW, THEREFORE, in consideration of the mutual covenants and agreements
herein contained, and intending to be legally bound hereby, the Company and
Buyers hereby agree as follows:

                                   ARTICLE I.

                            TERMS OF THE TRANSACTION

         1.1 Agreement to Issue Shares. Within five (5) business days of
Closing, and on the terms and subject to the conditions set forth in this
Agreement, the Company shall have issued to each Buyer, and each Buyer
(severally) shall purchase and accept from the Company, the number of shares of
common stock of the Company (the "Common Stock") as set forth beside its name on
Schedule I (as issued to all Buyers, collectively, the "Shares"); provided,
however, that in the case of Aspect and Special Situations, the Company shall
issue each such Buyer's shares concurrent with its delivery of such Buyer's
share of the Purchase Price to the Company, which shall occur within thirty (30)
days after Closing.

         1.2 Agreement to Issue Warrants . Within five (5) business days of
Closing, and on the terms and subject to the conditions set forth in this
Agreement, the Company shall issue and deliver to each Buyer the number of
warrants to purchase common stock of the Company as set forth beside its name on
Schedule I hereto (as issued to all Buyers, collectively, the "Warrants");
provided, however, that in the case of Aspect and Special Situations, the
Company shall issue such Buyer's warrants concurrent with such Buyer's delivery
of its share of the Purchase Price to the Company, which shall occur within
thirty (30) days after Closing.

         1.3 Purchase Price and Payment. In consideration of the sale of the
Shares and the Warrants, contemporaneously with its receipt of its portion of
the Shares and Warrants, each Buyer shall pay to the Company, the purchase price
set forth beside its name on Schedule I, the aggregate of which shall be the
"Purchase Price", the Purchase Price per Share (the "Issuance Price") was
determined by reference to the greater of (a) $2.05, and (b) the minimum Average
Price per Share necessary to prevent triggering antidilution provisions in any
agreement in existence as of the date hereof. Each Buyer shall pay its portion
of the Purchase Price to the Company in immediately available funds by confirmed
wire transfer to a bank account to be designated by the Company. Special
Situations and Aspect shall have a period of thirty (30) days after the Closing
Date to fund their respective allocable

                                       1
<PAGE>   3

shares of the Purchase Price; if Aspect fails to so timely fund its share of the
Purchase Price, Special Situations shall have no obligation to do so and this
Agreement shall automatically, unless Special Situations nonetheless voluntarily
funds its share of the Purchase Price within such 30-day period, have no further
force and effect relative to Special Situations. Nothing in this Section 1.3 or
elsewhere in this Agreement shall be construed to change or diminish Aspect's
absolute obligation to fund its share of the Purchase Price within 30 days after
the Closing Date.

         1.4 Limitations. The rights and interests afforded Buyers hereunder
and/or under the Warrant Certificate are subject to any pre-existing rights of
the Company's common stock, option and warrant holders.

                                   ARTICLE II.

                                     CLOSING

         The closing of the transactions contemplated hereby (the "Closing")
shall take place (i) at the offices of Brigham Exploration Company, 6300 Bridge
Point Parkway, Building 2, Suite 500, Austin, Texas, 78730, at 10:00 a.m., local
time, on February 22, 2000, or (ii) at such other time or place or on such other
date as the parties hereto shall agree. The date on which the Closing is
required to take place is herein referred to as the "Closing Date." All Closing
transactions shall, except as provided otherwise herein, be deemed to have
occurred simultaneously.

                                  ARTICLE III.

                  REPRESENTATIONS AND WARRANTIES OF THE COMPANY

         The Company represents and warrants to Buyers that:

         3.1 Corporate Organization. The Company is a corporation duly
organized, validly existing, and in good standing under the laws of Delaware and
has all requisite corporate power and corporate authority to own, lease, and
operate its properties and to carry on its business as now being conducted. No
actions or proceedings to dissolve the Company are pending or, to the best
knowledge of the Company, threatened.

         3.2 Qualification. The Company is duly qualified or licensed to do
business as a foreign corporation and is in good standing in each of the
jurisdictions in which such qualification or licensing is required for the
conduct of its business.

         3.3 Charter and Bylaws. Neither the Company nor any Subsidiary is in
violation of any provision of its charter or bylaws, other than violations
which, individually or in the aggregate, do not and will not have a Material
Adverse Effect on the Company.

         3.4 Capitalization of the Company. The authorized capital stock of the
Company consists of (a) 30,000,000 shares of common stock and, after giving
effect to the transactions contemplated herein and all other issuances of
capital stock of the Company on or prior to the date hereof, 16,712,908 shares
of common stock will be issued and outstanding and 731,707 shares will be

                                       2
<PAGE>   4

reserved for future issuance pursuant hereto, and (b) 10,000,000 shares of
preferred stock, none of which are presently outstanding. Schedule III sets
forth a description of the outstanding capital stock of the Company, including
options and warrants.

         3.5 Authority Relative to This Agreement. The Company has full
corporate power and corporate authority to execute, deliver, and perform this
Agreement and the Ancillary Documents (as hereinafter defined) to which it is a
party and to consummate the transactions contemplated hereby and thereby. The
execution, delivery, and performance by the Company of this Agreement and the
Ancillary Documents to which it is a party, and the consummation by it of the
transactions contemplated hereby and thereby, have been duly authorized by all
necessary corporate action of the Company. This Agreement has been duly executed
and delivered by the Company and constitutes, and each Ancillary Document
executed or to be executed by the Company has been, or when executed will be,
duly executed and delivered by the Company and constitute, or when executed and
delivered will constitute, valid and legally binding obligations of the Company,
enforceable against the Company in accordance with their respective terms,
except that such enforceability may be limited by (i) applicable bankruptcy,
insolvency, reorganization, moratorium, and similar laws affecting creditors'
rights generally and (ii) equitable principles which may limit the availability
of certain equitable remedies (such as specific performance) in certain
instances.

         3.6 Shares and Warrants. The Shares to be issued by the Company within
five (5) business days of Closing or, in the case of Special Situations and
Aspect, within thirty (30) days after Closing, and the Shares issuable upon
exercise of the Warrants (the "Warrant Shares") have been duly authorized for
such issuance. When issued and delivered by the Company in accordance with the
provisions of this Agreement, the Shares, the Warrants and the Warrant Shares
will be validly issued, fully paid, and nonassessable. The issuance of the
Shares, the Warrants and the Warrant Shares pursuant to this Agreement is not
subject to any preemptive or similar rights.

         3.7 Approvals. Except for registration of the Shares under the
Securities Act of 1933, as amended (the "Securities Act"), and/or the rules and
regulations of the Securities and Exchange Commission (the "Commission")
promulgated thereunder, there, no authorizations, approvals or consents of, and
no filings or registrations with, any governmental authority are necessary for
the execution, delivery or performance by the Company of this Agreement or for
the validity or enforceability thereof.

         3.8 Disclosure. No representation or warranty made by the Company in
this Agreement, and no statement of the Company contained in (a) any document
filed by the Company with the Commission pursuant to the Company's reporting
obligations under the Securities Act and the Securities Exchange Act of 1934, as
amended (the "Exchange Act"), or (b) any document, certificate, or other writing
furnished or to be furnished by the Company pursuant hereto or in connection
herewith, contains or will contain, at the time of delivery, any untrue
statement of a material fact or omits or will omit to state any material fact
necessary in order to make the statements contained therein, in light of the
circumstances under which they are made, not misleading. The Company knows of no
matter (other than matters of a general character not relating solely to the
Company or any subsidiary in any specific manner) which has not been disclosed
to Buyer pursuant to this Agreement which has or, so far as the Company can now
reasonably foresee, will have a Material Adverse Effect. The Company has
complied in all material respects with its obligations under the Securities Act
and the Exchange Act.

                                       3
<PAGE>   5

         3.9 Representations and Warranties on Closing Date. The representations
and warranties made in this Article III will be true and correct in all material
respects on and as of the Closing Date with the same force and effect as if such
representations and warranties had been made on and as of the Closing Date,
except that any such representations and warranties which expressly relate only
to an earlier date shall be true and correct on the Closing Date as of such
earlier date.

                                   ARTICLE IV.

                    REPRESENTATIONS AND WARRANTIES OF BUYERS

         Each Buyer, only with respect to itself, represents and warrants to the
Company that:

         4.1 Organization and Formation. Each corporate Buyer is duly organized,
validly existing, and in good standing under the laws of the jurisdiction of its
incorporation and has all requisite corporate power and corporate authority to
own, lease, and operate its properties and to carry on its business as now being
conducted. Each partnership Buyer is duly formed and is in good standing (as
applicable) under the laws of the jurisdiction of its formation. No actions or
proceedings to dissolve any Buyer are pending or, to the best knowledge of
Buyers, threatened.

         4.2 Authority Relative to This Agreement. Each Buyer has full power and
authority to execute, deliver, and perform this Agreement and the Ancillary
Documents to which it is a party and to consummate the transactions contemplated
hereby and thereby. The execution, delivery, and performance by Buyers of this
Agreement and the Ancillary Documents to which they are parties, and the
consummation by them of the transactions contemplated hereby and thereby, have
been duly authorized by all necessary corporate or partnership action, as
applicable, of Buyers. This Agreement has been duly executed and delivered by
Buyers and constitutes, and each Ancillary Document executed or to be executed
by Buyers has been, or when executed will be, duly executed and delivered by
each Buyer and constitute, or when executed and delivered will constitute, valid
and legally binding obligations of each Buyer, enforceable against each Buyer in
accordance with their respective terms, except that such enforceability may be
limited by (i) applicable bankruptcy, insolvency, reorganization, moratorium,
and similar laws affecting creditors' rights generally and (ii) equitable
principles which may limit the availability of certain equitable remedies (such
as specific performance) in certain instances.

         4.3 Financing. Each Buyer has, and at the Closing will have, such funds
as are necessary for the consummation by it of the transactions contemplated
hereby.

         4.4 Disclosure of Information. Buyers represent that they have had an
opportunity to ask questions of and receive answers from the Company regarding
the Company and its business, assets, results of operation, and financial
condition and the terms and conditions of the issuance of the Shares. Each Buyer
further represents that it has access to all filings duly made by the Company
with the Securities and Exchange Commission.

         4.5 Investment Experience. Each Buyer acknowledges that it can bear the
economic risk of

                                       4
<PAGE>   6

its investment in the Shares, and has such knowledge and experience in financial
and business matters that it is capable of evaluating the merits and risks of an
investment in the Shares.

         4.6 Restricted Securities. Each Buyer understands that the Shares will
not have been registered pursuant to the Securities Act or any applicable state
securities laws, that the Shares will be characterized as "restricted
securities" under federal securities laws, and that under such laws and
applicable regulations the Shares cannot be sold or otherwise disposed of
without registration under the Securities Act or an exemption therefrom. In this
connection, each Buyer represents that it is familiar with Rule 144 promulgated
under the Securities Act, as currently in effect, and understands the resale
limitations imposed thereby and by the Securities Act. Appropriate stop transfer
instructions may be issued to the transfer agent for securities of the Company
(or a notation may be made in the appropriate records of the Company) in
connection with the Shares.

         4.7 Accredited Investor; Investment Intent. Each Buyer is an accredited
investor as defined in Regulation D under the Securities Act. Each Buyer is
acquiring its portion of the Shares for its own account for investment and not
with a view to, or for sale or other disposition in connection with, any
distribution of all or any part thereof, except in compliance with applicable
federal and state securities laws.

         4.8 Representations and Warranties on Closing Date. The representations
and warranties made in this Article IV will be true and correct in all material
respects on and as of the Closing Date with the same force and effect as if such
representations and warranties had been made on and as of the Closing Date,
except that any such representations and warranties which expressly relate only
to an earlier date shall be true and correct on the Closing Date as of such
earlier date.

                                   ARTICLE V.

                              ADDITIONAL AGREEMENTS

         5.1 Reasonable Best Efforts. Each party hereto agrees that it will not
voluntarily undertake any course of action inconsistent with the provisions or
intent of this Agreement and will use its reasonable best efforts to take, or
cause to be taken, all action and to do, or cause to be done, all things
reasonably necessary, proper, or advisable under Applicable Laws to consummate
the transactions contemplated by this Agreement, including, without limitation,
(i) cooperation in determining whether any consents, approvals, orders,
authorizations, waivers, declarations, filings, or registrations of or with any
Governmental Entity or third party are required in connection with the
consummation of the transactions contemplated hereby; (ii) reasonable best
efforts to obtain any such consents, approvals, orders, authorizations, and
waivers and to effect any such declarations, filings, and registrations; (iii)
reasonable best efforts to cause to be lifted or rescinded any injunction or
restraining order or other order adversely affecting the ability of the parties
to consummate the transactions contemplated hereby; (iv) reasonable best efforts
to defend, and cooperation in defending, all lawsuits or other legal proceedings
challenging this Agreement or the consummation of the transactions contemplated
hereby; and (v) the execution of any additional instruments necessary to
consummate the transactions contemplated hereby.

                                       5
<PAGE>   7

         5.2 Public Announcements. Except as may be required by Applicable Law
or this Section 5.2, no Buyer shall issue any press release or otherwise make
any public statement with respect to this Agreement or the transactions
contemplated hereby without the prior written consent of the other parties
(which consent shall not be unreasonably withheld). The Company shall prepare
and issue any press release or public statement required by Applicable Law.

         5.3 Notice of Litigation. Until the Closing, (i) Buyers, upon learning
of the same, shall promptly notify the Company of any Proceeding which is
commenced or threatened against a Buyer and which affects this Agreement or the
transactions contemplated hereby and (ii) the Company, upon learning of the
same, shall promptly notify Buyers of any Proceeding which is commenced or
threatened against the Company and which affects this Agreement or the
transactions contemplated hereby and any Proceeding which is commenced or
threatened against the Company or any Subsidiary.

         5.4 Fees and Expenses. The Company shall be responsible for the payment
of all of the Company's fees and expenses incurred in connection with this
Agreement and the transactions contemplated hereby and, except as expressly
provided otherwise in Section 9.7 hereof, the Buyers shall be responsible for
the payment of all the Buyers' fees and expenses incurred in connection with
this Agreement and the transactions contemplated hereby.

         5.5 Transfer Taxes. All sales, transfer, filing, recordation,
registration, stamp, and similar Taxes and fees arising from or associated with
the sale and transfer of the Shares as contemplated hereunder shall be borne by
the Buyers.

         5.6 Survival of Covenants. Except for any covenant or agreement, which
by its terms expressly terminates as of a specific date or event, the covenants
and agreements of the parties hereto contained in this Agreement shall survive
the Closing without contractual limitation.

                                   ARTICLE VI.

                    CONDITIONS TO OBLIGATIONS OF THE COMPANY

         The obligations of the Company to consummate the transactions
contemplated by this Agreement shall be subject to the fulfillment on or prior
to the Closing Date of each of the following conditions:

         6.1 Representations and Warranties True. All the representations and
warranties of Buyers contained in this Agreement, and in any agreement,
instrument, or document delivered pursuant hereto or in connection herewith on
or prior to the Closing Date, shall be true and correct in all material respects
as of the date made and (having been deemed to have been made again on and as of
the Closing Date in the same language) shall be true and correct in all material
respects on and as of the Closing Date, except as affected by transactions
permitted by this Agreement and except to the extent that any such
representation or warranty is made as of a specified date, in which case such
representation or warranty shall have been true and correct in all material
respects as of such specified date.

                                       6
<PAGE>   8

         6.2 Covenants and Agreements Performed. Buyers shall have performed and
complied with in all material respects all covenants and agreements required by
this Agreement to be performed or complied with by them on or prior to the
Closing Date.

         6.3 Legal Proceedings. No Proceeding shall, on the Closing Date, be
pending or threatened seeking to restrain, prohibit, or obtain damages or other
relief in connection with this Agreement or the consummation of the transactions
contemplated hereby.

         6.4 Consents.

         (a) There shall have been obtained any and all material permits,
consents, and approvals of Governmental Entities that reasonably may be deemed
necessary so that the consummation of the transactions contemplated hereby will
be in compliance with Applicable Law, the failure to comply with which would
have a Material Adverse Effect on the Company.

         (b) All consents and approvals of private persons, (i) the granting of
which is necessary for the consummation of the transactions contemplated hereby
and (ii) the non-receipt of which would have a Material Adverse Effect on the
Company, shall have been obtained.

                                   ARTICLE VII

                       CONDITIONS TO OBLIGATIONS OF BUYERS

         The obligations of Buyers to consummate the transactions contemplated
by this Agreement shall be subject to the fulfillment on or prior to the Closing
Date of each of the following conditions:

         7.1 Representations and Warranties True. All the representations and
warranties of the Company contained in this Agreement, and in any agreement,
instrument, or document delivered pursuant hereto or in connection herewith on
or prior to the Closing Date, shall be true and correct in all material respects
(except for such representations and warranties that are limited by materiality,
in which case they shall be true and correct in all respects) as of the date
made and (having been deemed to have been made again on and as of the Closing
Date in the same language) shall be true and correct in all material respects on
and as of the Closing Date, except as affected by transactions permitted by this
Agreement and except to the extent that any such representation or warranty is
made as of a specified date, in which case such representation or warranty shall
have been true and correct in all material respects as of such specified date.

         7.2 Covenants and Agreements Performed. The Company shall have
performed and complied with in all material respects all covenants and
agreements required by this Agreement to be performed or complied with by it on
or prior to the Closing Date.

         7.3 Legal Proceedings. No Proceeding shall, on the Closing Date, be
pending or threatened seeking to restrain, prohibit, or obtain damages or other
relief in connection with this Agreement or the consummation of the transactions
contemplated hereby.

                                       7
<PAGE>   9

         7.4 Ancillary Documents. Buyers shall have received the certificates,
instruments, and documents listed below (the "Ancillary Documents"):

                  (a) In accordance with the denominations designated in
         Schedule I, stock certificates in definitive form and duly executed on
         behalf of the Company, representing the portion of the Shares
         registered in the name of each Buyer.

                  (b) In accordance with the denominations designed in Schedule
         I, Warrant Certificates in substantially the form attached hereto as
         Schedule II.

                  (c) A copy of the resolutions of the Board of Directors of the
         Company authorizing the execution, delivery, and performance by the
         Company of this Agreement, certified by the secretary or an assistant
         secretary of the Company.

                  (d) Such other certificates, instruments, and documents as may
         be reasonably requested by Buyers to carry out the intent and purposes
         of this Agreement.

         7.5 Special Conditions Precedent. As a precondition to the obligations
of Buyers, (a) the Company shall have entered, or shall have caused Brigham Oil
& Gas, L.P. ("BOG") to enter, into an Amended and Restated Credit Agreement with
its senior lenders affording the Company and/or BOG additional availability up
to, but not exceeding total availability of $15,000,000, and (b) the total
Purchase Price for all shares issued to the Buyers hereunder equals or exceeds
$4,500,000.

                                  ARTICLE VIII.

                       TERMINATION, AMENDMENT, AND WAIVER

         8.1 Termination. This Agreement may be terminated and the transactions
contemplated hereby abandoned at any time prior to the Closing in the following
manner:

               (a)  by mutual written consent of the Company and Buyers; or

               (b)  by either the Company or Buyers, if:

                    (i) the Closing shall not have occurred on or before
                    February 23, 2000, unless such failure to close shall be due
                    to a breach of this Agreement by the party seeking to
                    terminate this Agreement pursuant to this clause (i); or

                    (ii) there shall be any statute, rule, or regulation that
                    makes consummation of the transactions contemplated hereby
                    illegal or otherwise prohibited or a Governmental Entity
                    shall have issued an order, decree, or ruling or taken any
                    other action permanently restraining, enjoining, or
                    otherwise prohibiting the consummation of the transactions
                    contemplated hereby, and such order, decree, ruling, or
                    other action shall have become final and nonappealable; or

               (c)  by the Company, if (i) any of the representations and
                    warranties of Buyers

                                       8
<PAGE>   10

                    contained in this Agreement shall not be true and correct in
                    any respect which is material to Buyers or the ability of
                    Buyers to consummate the transactions contemplated hereby,
                    or (ii) Buyers shall have failed to fulfill in any material
                    respect any of their obligations under this Agreement, and,
                    in the case of each of clauses (i) and (ii), such
                    misrepresentation, breach of warranty, or failure (provided
                    it can be cured) has not been cured within ten days after
                    written notice thereof from the Company to Buyers; or

               (d)  by Buyers, if (i) any of the representations and warranties
                    of the Company contained in this Agreement shall not be true
                    and correct in any respect which is material to the Company
                    and the Subsidiaries considered as a whole or the ability of
                    the Company to consummate the transactions contemplated
                    hereby, or (ii) the Company shall have failed to fulfill in
                    any material respect any of its obligations under this
                    Agreement, and, in the case of each of clauses (i) and (ii),
                    such misrepresentation, breach of warranty, or failure
                    (provided it can be cured) has not been cured within ten
                    days after written notice thereof from Buyers to the
                    Company.

         8.2 Effect of Termination. In the event of the termination of this
Agreement pursuant to Section 8.1 by the Company, on the one hand, or Buyer, on
the other, written notice thereof shall forthwith be given to the other party
specifying the provision hereof pursuant to which such termination is made, and
this Agreement shall become void and have no effect, except that the agreements
contained in this Section and in Sections 5.2, 5.4, and 11.1 shall survive the
termination hereof. Nothing contained in this Section shall relieve any party
from liability for damages actually incurred as a result of any breach of this
Agreement.

         8.3 Amendment. This Agreement may not be amended except by an
instrument in writing signed by or on behalf of all the parties hereto.

         8.4 Waiver. The Company, on the one hand, or Buyers, on the other, may:
(i) waive any inaccuracies in the representations and warranties of the other
contained herein or in any document, certificate, or writing delivered pursuant
hereto, or (ii) waive compliance by the other with any of the other's agreements
or fulfillment of any conditions to its own obligations contained herein. Any
agreement on the part of a party hereto to any such waiver shall be valid only
if set forth in an instrument in writing signed by or on behalf of such party.
No failure or delay by a party hereto in exercising any right, power, or
privilege hereunder shall operate as a waiver thereof nor shall any single or
partial exercise thereof preclude any other or further exercise thereof or the
exercise of any other right, power, or privilege.

         8.5 Remedies Not Exclusive. The rights and remedies herein provided
shall be cumulative and not exclusive of any rights or remedies provided by law.
The rights and remedies of any party based upon, arising out of, or otherwise in
respect of any inaccuracy in or breach of any representation, warranty,
covenant, or agreement contained in this Agreement shall in no way be limited by
the fact that the act, omission, occurrence, or other state of facts upon which
any claim of any such inaccuracy or breach is based may also be the subject
matter of any other representation, warranty, covenant, or agreement contained
in this Agreement (or in any other agreement between the parties) as to which
there is no inaccuracy or breach.

                                       9
<PAGE>   11

                                   ARTICLE IX

              RESTRICTIONS ON TRANSFERABILITY; REGISTRATION RIGHTS

         9.1 Transfers.

         (a) General. The Shares and the Warrants shall not be transferred
before satisfaction of the conditions specified in this Article IX, which
conditions are intended to reflect the intent of the Company and Buyers and
further to ensure compliance with the provisions of the Securities Act and
applicable state securities laws. Each Buyer, by entering into this Agreement
and accepting the Shares and the Warrants, agrees to be bound by the provisions
of this Article IX.

         (b) Restrictive Legend. Except as otherwise provided in this Article
IX, each certificate representing Warrants, Shares or Warrant Shares shall be
stamped or otherwise imprinted with a legend in substantially the following
form:

         "THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN
         REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR ANY
         APPLICABLE STATE SECURITIES LAWS, AND MAY NOT BE SOLD OR TRANSFERRED IN
         THE ABSENCE OF SUCH REGISTRATION OR AN EXEMPTION THEREFROM. SUCH
         SECURITIES ARE SUBJECT TO THE RESTRICTIONS AND PRIVILEGES SPECIFIED IN
         A SECURITIES PURCHASE AND REGISTRATION RIGHTS AGREEMENT, DATED AS OF
         FEBRUARY 22, 2000, BETWEEN BRIGHAM EXPLORATION COMPANY AND THE INITIAL
         HOLDERS OF SECURITIES NAMED THEREIN, A COPY OF WHICH IS ON FILE WITH
         THE SECRETARY OF BRIGHAM EXPLORATION COMPANY AND WILL BE FURNISHED
         WITHOUT CHARGE TO THE HOLDER HEREOF UPON WRITTEN REQUEST, AND THE
         HOLDER OF THIS CERTIFICATE AGREES TO BE BOUND THEREBY."

         9.2 Registration. The Company shall utilize reasonable best efforts to,
within 90 days after the Closing Date, register on a Form S-3 (or Form S-1 or
successor form to either) Registration Statement all of the Registrable
Securities that are held by any Buyer(s) that, on or before the Closing Date,
have elected to have their Registrable Securities so registered. The Company
shall use its reasonable best efforts to keep such Registration Statement
effective until the twenty-four (24) month anniversary of such Effective Date.

         9.3 Incidental Registration.

         If the Company at any time proposes to file on its behalf and/or on
behalf of any of its security holders (the "Demanding Security Holders") a
Registration Statement under the Securities Act on any form (other than a
Registration Statement (i) filed pursuant to demand under the Company's
Registration Rights Agreement with Joint Energy Development Investments II
Limited Partnership, a Delaware limited partnership, and Enron Capital & Trade
Resources Corp., a Delaware corporation, dated August 20, 1998, as amended, or
(ii) on Form S-8 or any similar or successor form or any other registration
statement relating to an offering of securities solely to the Company's

                                       10
<PAGE>   12

existing security holders or employees) to register the offer and sale of its
common stock for cash, it will give written notice to all Buyers at least five
(5) days before the anticipated date of initial filing with the Commission of
such Registration Statement, which notice shall set forth the Company's
intention to effect such a registration, the class or series and number of
equity securities proposed to be registered and the intended method of
disposition of the securities proposed to be registered by the Issuer. The
notice shall offer to include in such filing all of each Buyer's Registrable
Securities.

         Each Buyer desiring to have Registrable Securities registered under
this Section 9.3 shall advise the Company in writing within fifteen (15) days
after the date of receipt of such offer from the Company, setting forth the
amount of such Registrable Securities for which registration is requested. The
Company shall thereupon include in such filing the number of Registrable
Securities for which registration is so requested, subject to the next sentence,
and shall use its best efforts to effect registration under the Securities Act
of such securities. If the managing underwriter of a proposed public offering
shall advise the Company in writing that, in its opinion, the distribution of
the Registrable Securities requested to be included in the registration
concurrently with the securities being registered by the Company or the
Demanding Security Holder, as the case may be, would materially and adversely
affect the distribution of such securities by the Company or such Demanding
Security Holders, as the case may be, then all selling security holders (but not
the Company or such Demanding Security Holders, as the case may be) shall reduce
the amount of securities each intended to distribute through such offering on a
pro rata basis to the greatest aggregate amount which, in the opinion of such
managing underwriter, would not materially and adversely affect the distribution
of such securities.

         Nothing in this Section 9.3 shall preclude the Company from
discontinuing the registration of its securities being effected on its behalf
under this Section 9.3 at any time prior to the effective date of the
registration relating thereto. Notwithstanding any provision herein, the rights
of the Buyers under this Section 9.3 are subject to the express limitations
contained in registration rights agreements (and registration rights provisions
in agreements) in effect on the date hereof between the Company and other
parties.

         9.4 Requests for Registration. Each Buyer agrees that it will not sell,
transfer or otherwise dispose of any Registrable Securities in whole or part,
except pursuant to an effective Registration Statement under the Securities Act
or an exemption from registration thereunder. Each certificate, if any,
evidencing such Registrable Securities issued upon such transfer shall bear the
restrictive legend set forth in Section 9.1, and each Warrant Certificate issued
upon such transfer shall bear the restrictive legend set forth in Section 9.1 of
this Agreement, unless in the opinion of the transferee's or Buyer's reputable
counsel delivered to the Company in connection with such transfer, such legend
is not required in order to ensure compliance with the Securities Act.

         9.5 Registration Procedures. In its efforts to effect registration of
the Registrable Securities pursuant to Section 9.2 or 9.3, above, the Company
will, as expeditiously as possible:

         (a) furnish to the selling security holders such number of copies or a
summary prospectus or other prospectus, including a preliminary prospectus, in
conformity with the requirements of the Securities Act, and such other
documents, as such selling security holders may reasonably request;

                                       11
<PAGE>   13

         (b) use its best efforts to register or qualify the securities covered
by such Registration Statement under such other securities or blue sky laws of
such jurisdictions within the United States as each holder of such securities
shall request (provided, however, the Company shall not be obligated to qualify
as a foreign corporation to do business under the laws of any jurisdiction in
which it is not then qualified or to file any general consent to service or
process), and do such other reasonable acts and things as may be requited of it
to enable such holder to consummate the disposition in such jurisdiction of the
securities covered by such Registration Statement;

         (c) enter into customary agreements (including an underwriting
agreement in customary form) and take such other actions as are reasonably
required in order to expedite or facilitate the disposition of such Shares; and

         (d) otherwise use its best efforts to comply with all applicable rules
and regulations of the Commission, and make available to its security holders,
as soon as reasonably practicable, but not later than 18 months after the
effective date of the Registration Statement, an earnings statement covering the
period of at least 12 months beginning with the first full month after the
effective date of such Registration Statement, which earnings statements shall
satisfy the provisions of Section 11(a) of the Securities Act.

         It shall be a condition precedent to the obligation of the company to
take any action pursuant to this Article IX in respect of the Registrable
Securities which are to be registered at the request of any Buyer that such
Buyer shall refurnish to the Company such information regarding the Registrable
Securities held by such Buyer and the intended method of disposition thereof as
the Company shall reasonably request and as shall be required in connection with
the action taken by the Company.

         9.6 Rights of Others.

         (a) The Company and any party to any existing agreements with the
Company who are entitled to piggy-back registration rights ("Other Holders")
with respect to a Registration Statement filed pursuant to Section 9.2 or 9.3
may include securities of the Company in such Registration Statement.

         (b) Notwithstanding any provision herein, the rights of the Buyers
under this Article IX are subject to the express limitations contained in
registration rights agreements (and registration rights provisions in
agreements) in effect on the date hereof between the Company and other parties.

         9.7 Expenses. All registration and filing fees (including those
incident to filing with the NASD), incurred in complying with this Article IX
(all of such fees shall be collectively referred to herein as "Registration
Expenses"), shall be paid by the Company; provided, however, the Company shall
not be responsible for any attorneys' fees or discount or commission or cost
reimbursement to any underwriter in respect of the securities sold by such
Buyer, or any other expenses incurred by Buyers in relation to this Agreement,
except as set forth in this Section 9.7.

                                       12
<PAGE>   14

         9.8 Liquidated Damages. The parties hereto agree that, without
limitation of any other remedies that may be available to Buyers, upon the
failure of the Company to register the shares of Registrable Securities that are
held by Buyers that have elected on or before the Closing Date to have their
Registrable Securities registered, within 180 days of the Closing Date, the
Buyers that have so elected to have their Registrable Securities registered
shall each be entitled to liquidated damages from the Company in an amount equal
to 1% per month of the Purchase Price paid by such Buyer, with the total
liquidated damages payable by the Company to all Buyers together not to exceed
$100,000. The parties agree that the liquidated damages set forth herein are
reasonable and not a penalty because of the difficulty of ascertaining the
actual damages that would be sustained by the Buyers in the event of such a
material breach of this Agreement by the Company.

                                   ARTICLE X.

                                  MISCELLANEOUS

         10.1 Notices. All notices, requests, demands, and other communications
required or permitted to be given or made hereunder by any party hereto shall be
in writing and shall be deemed to have been duly given or made if (i) delivered
personally, (ii) transmitted by first class registered or certified mail,
postage prepaid, return receipt requested, (iii) sent by prepaid overnight
courier service, or (iv) sent by telecopy or facsimile transmission, answer back
requested, to the parties at the following addresses (or at such other addresses
as shall be specified by the parties by like notice):

              If to:  GAP Coinvestment Partners II, L.P.
                      c/o General Atlantic Service Corporation
                      3 Pickwick Plaza
                      Greenwich, CT  06830
                      Attention: Mr. Steve Reynolds
                      Telefax:   (203) 622-8818

              with a copy to:
                      Paul, Weiss, Rifkind, Wharton & Garrison
                      1285 Avenue of the Americas
                      New York, NY  10019-6064
                      Attention: Douglas A. Cifu, Esq.
                      Telefax:   (212) 757-3990

              If to:  Special Situations Private Equity Fund
                      153 East 53rd Street
                      New York, NY  10022
                      Attention: Mr. Steven R. Becker
                      Telefax:   (212) 207-6515

              If to:  Aspect Resources, L.L.C.
                      511 16th Street, Suite 300
                      Denver, CO  80202
                      Attention: Mr. Alex Cranberg
                      Telefax:   (303) 573-7340

                                       13
<PAGE>   15

              If to the Company:

                      Brigham Exploration Company
                      6300 Bridge Point Parkway
                      Building 2, Suite 500
                      Austin, TX  78730
                      Attention: Chief Financial Officer
                      Telefax:   (512) 427-3400

Such notices, requests, demands, and other communications shall be effective (i)
if delivered personally or sent by courier service, upon actual receipt by the
intended recipient, (ii) if mailed, upon the earlier of five days after deposit
in the mail or the date of delivery as shown by the return receipt therefor, or
(iii) if sent by telecopy or facsimile transmission, when the answer back is
received.

         10.2 Entire Agreement. This Agreement, together with the Schedules,
Exhibits, Annexes, and other writings referred to herein or delivered pursuant
hereto, constitute the entire agreement between the parties hereto with respect
to the subject matter hereof and supersede all prior agreements and
understandings, both written and oral, between the parties with respect to the
subject matter hereof.

         10.3 Binding Effect; Assignment; No Third Party Benefit. This Agreement
shall be binding upon and inure to the benefit of the parties hereto and their
respective heirs, legal representatives, successors, and permitted assigns.
Except as otherwise expressly provided in this Agreement, neither this Agreement
nor any of the rights, interests, or obligations hereunder shall be assigned by
any of the parties hereto without the prior written consent of the other
parties. Nothing in this Agreement, express or implied, is intended to or shall
confer upon any person other than the parties hereto, and their respective
heirs, legal representatives, successors, and permitted assigns, any rights,
benefits, or remedies of any nature whatsoever under or by reason of this
Agreement.

         10.4 Severability. If any provision of this Agreement is held to be
unenforceable, this Agreement shall be considered divisible and such provision
shall be deemed inoperative to the extent it is deemed unenforceable, and in all
other respects this Agreement shall remain in full force and effect; provided,
however, that if any such provision may be made enforceable by limitation
thereof, then such provision shall be deemed to be so limited and shall be
enforceable to the maximum extent permitted by Applicable Law.

         10.5 GOVERNING LAW. THIS AGREEMENT SHALL BE GOVERNED BY AND CONSTRUED
AND ENFORCED IN ACCORDANCE WITH THE INTERNAL LAWS OF THE STATE OF TEXAS, WITHOUT
REGARD TO THE PRINCIPLES OF CONFLICTS OF LAWS THEREOF.

         10.6 Counterparts. This Agreement may be executed by the parties hereto
in any number of counterparts, each of which shall be deemed an original, but
all of which shall constitute one and the same agreement. Each counterpart may
consist of a number of copies hereof each signed by less than all, but together
signed by all, the parties hereto.

                                       14
<PAGE>   16

         10.7 Consent to Jurisdiction

         (a) The parties hereto hereby irrevocably submit to the jurisdiction of
the courts of the State of Texas and the federal courts of the United States of
America located in Dallas, Texas, and appropriate appellate courts therefrom,
over any dispute arising out of or relating to this Agreement or any of the
transactions contemplated hereby, and each party hereby irrevocably agrees that
all claims in respect of such dispute or proceeding may be heard and determined
in such courts.

         (b) Each of the parties hereto hereby consents to process being served
by any party to this Agreement in any suit, action, or proceeding of the nature
specified in subsection (a) above by the mailing of a copy thereof in the manner
specified by the provisions of Section 10.1.

         10.8 Liability of Buyers. The liability of each Buyer with respect to
the agreements, covenants, representations and warranties of Buyers contained in
this Agreement or in any certificate, instrument, or document delivered pursuant
hereto shall be to the extent such agreements, covenants, representations or
warranties applies to himself, herself, or itself and not with respect to any
other Buyer.

         10.9 Consent to Certain Stock Issuances. By its execution hereof, each
Buyer hereby consents to the issuance of Common Stock upon exercise of warrants
and options to purchase Common Stock outstanding as of the date hereof.

         10.10 Loss or Mutilation. Upon receipt by the Company from any Buyer of
evidence reasonably satisfactory to it of the ownership of and the loss, theft,
destruction or mutilation of a certificate representing Shares and indemnity
reasonably satisfactory to it (it being understood that the written agreement of
the Buyer or an Affiliate thereof shall be sufficient indemnity) and in case of
mutilation upon surrender and cancellation hereof or thereof, the Company will
execute and deliver in lieu hereof or thereof a new stock certificate, of like
tenor to such Buyer, provided, in the case of mutilation, no indemnity shall be
required if the certificate representing Shares in identifiable form is
surrendered to the Company for cancellation.

                                   ARTICLE XI.

                                   DEFINITIONS

         11.1 Certain Defined Terms. As used in this Agreement, each of the
following terms has the meaning given it below:

                  "Affiliate" means, with respect to any person, any other
         person that, directly or indirectly, through one or more
         intermediaries, controls, is controlled by, or is under common control
         with, such person. For the purposes of this definition, "control" when
         used with respect to any person means the possession, directly or
         indirectly, of the power to direct or cause the direction of the
         management and policies of such person, whether through the ownership
         of voting securities, by contract, or otherwise; and the terms
         "controlling" and "controlled" have meanings correlative to the
         foregoing.

                                       15
<PAGE>   17

                  "Applicable Law" means any statute, law, rule, or regulation
         or any judgment, order, writ, injunction, or decree of any Governmental
         Entity to which a specified person or property is subject.

                  "Code" means the Internal Revenue Code of 1986, as amended.

                  "Exchange Act" means the Securities Exchange Act of 1934, as
         amended.

                  "Governmental Entity" means any court or tribunal in any
         jurisdiction (domestic or foreign) or any federal, state, municipal, or
         other governmental body, agency, authority, department, commission,
         board, bureau, or instrumentality (domestic or foreign), as well as the
         New York Stock Exchange, The NASDAQ Stock Market, and any exchange upon
         which the Common Stock is listed from time to time.

                  "Material Adverse Effect" means any change, development, or
         effect (individually or in the aggregate) which is, or is reasonably
         likely to be, materially adverse (i) to the business, assets, results
         of operations or condition (financial or otherwise) of a party, or (ii)
         to the ability of a party to perform on a timely basis any material
         obligation under this Agreement or any agreement, instrument, or
         document entered into or delivered in connection herewith.

                                       16
<PAGE>   18

                  "Person" means any individual, corporation, partnership, joint
         venture, association, joint-stock company, trust, enterprise,
         unincorporated organization, or Governmental Entity.

                  "Proceedings" means all proceedings, actions, claims, suits,
         investigations, and inquiries by or before any arbitrator or
         Governmental Entity.

                  "Registrable Securities" means the Shares, the Warrant Shares
         and any shares issued upon exercise of the Warrants.

                  "Securities Act" means the Securities Act of 1933, as amended.

         "Taxes" means any income taxes or similar assessments or any sales,
excise, occupation, use, ad valorem, property, production, severance,
transportation, employment, payroll, franchise, or other tax imposed by any
United States federal, state, or local (or any foreign or provincial) taxing
authority, including any interest, penalties, or additions attributable thereto.

     IN WITNESS WHEREOF, the parties have executed this Agreement, or caused
this Agreement to be executed by their duly authorized representatives, all as
of the day and year first above written.

                                              THE COMPANY:

                                              BRIGHAM EXPLORATION COMPANY

                                              By:
                                                 -------------------------------
                                                 Name:   Ben M. Brigham
                                                       -------------------------
                                                 Title:  President and CEO
                                                       -------------------------

                                              BUYERS:

                                              GAP COINVESTMENT PARTNERS II, L.P.

                                              By:
                                                 -------------------------------
                                                 Name:
                                                       -------------------------
                                                 Title:
                                                       -------------------------

                                       17
<PAGE>   19

                                              SPECIAL SITUATIONS PRIVATE
                                              EQUITY FUND, L.P.

                                              By:
                                                 -------------------------------
                                                 Name:
                                                       -------------------------
                                                 Title:
                                                       -------------------------

                                              ASPECT RESOURCES, L.L.C.
                                              By its Manager, Aspect Management
                                              Corporation

                                              By:
                                                 -------------------------------
                                                 Name:
                                                       -------------------------
                                                 Title:
                                                       -------------------------

                                       18
<PAGE>   20

                                   SCHEDULE I

<TABLE>
<CAPTION>

                                                                                Shares of              Purchase
               Buyers                                       Warrants           Common Stock             Price
              --------                                    -------------        -------------        -------------
<S>                                                       <C>                  <C>                  <C>
GAP Coinvestment Partners II, L.P.                              325,203              975,610        $2,000,000.00
                                                          -------------        -------------        -------------
Special Situations Private Equity Fund, L.P.                    243,902              731,707        $1,500,000.00
                                                          -------------        -------------        -------------
Aspect Resources, L.L.C                                         162,602              487,805        $1,000,000.00
                                                          -------------        -------------        -------------

                        Total                                   731,707            2,195,122        $4,500,000.00
                                                          =============        =============        =============
</TABLE>

<PAGE>   21

                                   SCHEDULE II

                               WARRANT CERTIFICATE

Number of Warrants: ____________                                Warrant No. ____

     This Warrant certificate ("Warrant Certificate") certifies that, for value
received,

----------------------------------------------------

is the registered holder of the number of warrants (the "Warrants") set forth
above. Each Warrant entitles the holder thereof, at any time to time during the
Exercise Period, to purchase from the Company one fully paid and nonassessable
share of Common Stock at the Exercise Price, subject to adjustment as provided
herein. Initially capitalized terms used but not defined herein shall have the
meanings ascribed to them in the Securities Purchase Agreement.

                  "Common Stock" means the common stock, $.01 per value per
         share, of the Company and such other class of securities as shall then
         represent the common equity of the Company.

                  "Company" means Brigham Exploration Company, a Delaware
         corporation.

                  "Exercise Period" means the period of time between the Closing
         Date, as defined in the Securities Purchase Agreement and 5:00 p.m.
         (New York City time) on the Expiration Date.

                  "Exercise Price," subject in all circumstances to adjustment
         in accordance with Section 2, means 125% of the Issuance Price.

                  "Expiration Date" means February 22, 2003.

                  "IPO" shall mean the initial public offering of securities of
         the Company consummated on May 24, 1997, pursuant to a registration
         statement filed under the Securities Act of 1933, as amended, and the
         rules and regulations promulgated thereunder.

                  "Issuance Date" means February 22, 2000.

                  "Person" means any individual, corporation, company,
         partnership, joint venture, trust, limited liability company,
         unincorporated organization or government or any agency,
         instrumentality or political subdivision thereof, or any other form of
         equity.

                  "Price" means the average of the "high" and "low" prices as
         reported in The Wall Street Journal's listing for such day (corrected
         for obvious typographical errors) or if such shares are not reported in
         such listing, the average of the reported "high" and "low" sales

                                    Sch II-1

<PAGE>   22

         prices on the largest national securities exchange (based on the
         aggregate dollar value of securities listed) on which such shares are
         listed or traded, or if such shares are not listed or traded on any
         national securities exchange, then the average of the reported "high"
         and "low" sales prices for such shares in the over-the-counter market,
         as reported on the National Association of Securities Dealers Automated
         Quotations System, or, if such prices shall not be reported thereon,
         the average of the closing bid and asked prices reported by the
         National Quotations Bureau Incorporated, or, in all other cases, the
         Estimated Private Market Equity Value divided by the number of
         outstanding shares (on a fully diluted basis using the treasury stock
         method). The "Average" Price per share for any period shall be
         determined by dividing the sum of the Prices determined for the
         individual trading days in such period by the number of trading days in
         such period.

                  "Publicly Traded" means, with respect to the Common Stock,
         that such securities are listed for trading on the New York Exchange,
         Inc., the American Stock Exchange, Inc. or the NASDAQ National Market.
         The Common Stock shall also be deemed to be Publicly Traded if the
         Common Stock is included in the NASDAQ SmallCap Market.

                  "Registrable Securities" means the Shares, the Warrant Shares
         and any shares issued upon exercise of the Warrants.

                  "Securities Purchase Agreement" means the Securities Purchase
         and Registration Agreement, dated as of February 22, 2000, among the
         Company and the Buyers.

         1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in whole or
in part, at any time or from time to time, during the Exercise Period, by (i)
presentation and surrender to the Company at its address set forth in Section 9
of this Warrant Certificate with the Election To Exercise, attached hereto as
Exhibit A, duly completed and executed, and (ii) payment of the Exercise Price,
by bank draft or cashier's check, for the number of Warrants being exercised. If
the holder of this Warrant Certificate at any time exercises less than all the
Warrants, the Company shall issue to such a holder a warrant certificate
identical in form to this Warrant Certificate, but evidencing a number of
Warrants equal to the number of Warrants originally represented and surrender of
this Warrant Certificate to the Company at its address set forth in Section 9
and at the request of the holder, the Company will, without expense, at the
option of the holder, issue to the holder in substitution for this Warrant
Certificate one or more warrant certificates in identical form and for an
aggregate number of Warrants equal to the number of Warrants evidenced by this
Warrant Certificate.

                (b) To the extent that the Warrants have not been exercised at
or prior to the Expiration Date, such Warrants expire and the rights of the
holder shall become void and of no effect.

                (c) In lieu of exercise in accordance with Section 1(a) above,
the Warrants represented hereby may be exercised, in whole or in part, by
presentation and surrender at the office of the Company specified herein of this
Warrant Certificate with the "Alternative Election to Exercise", attached hereto
as Exhibit A-1, duly completed and executed. Upon such exercise, the

                                    Sch II-2

<PAGE>   23

holder shall be entitled to receive from the Company, for each share of Common
Stock issuable upon exercise of each Warrant being exercised, shares of Common
Stock with a value equal to the amount by which either the Price or, in the
event that the Common Stock is not Publicly Traded, the Fair Market Value (as
defined below) of one share of Common Stock on the date of exercise exceeds the
Exercise Price (the "Appreciation Amount"). If the Common Stock is Publicly
Traded on the date of exercise of Warrants pursuant to this Section 1(c), the
Company shall deliver (or cause its transfer agent to deliver) certificates
representing shares of Common Stock issuable upon such exercise within ten
business days following the date of exercise. For purposes of this subsection,
in the event that the Common Stock is not Publicly Traded, the "Fair Market
Value" of the share of Common Stock shall be established by the Board of
Directors of the Company in its reasonable discretion. If the Common Stock is
not Publicly Traded on the date of exercise of Warrants pursuant to this Section
1(c), such election may be withdrawn by the holder at any time prior to the
third business day following the date on which the holder receives notice of the
final determination of the Fair Market Value of the share of Common Stock.
Unless the holder timely withdraws its exercise, the Company shall deliver (or
cause its transfer agent to deliver) certificates representing Common Stock so
issuable within 10 business days following the date of determination of the Fair
Market Value. In lieu of any fractional shares otherwise issuable upon exercise
pursuant hereto, the Company shall pay to the holder cash in an amount equal to
the Price, or in the event the Company is not Publicly Traded, the Fair Market
Value of such fractional share.

         2. ANTIDILUTION ADJUSTMENTS. The shares of Common Stock purchasable on
exercise of the Warrants are shares of Common Stock as constituted as of the
Issuance Date. The number and kind of securities purchasable upon the exercise
of the Warrants, and the Exercise Price, shall be subject to adjustment from
time to time upon the happening of certain events, as follows:

            Mergers, Consolidations and Reclassifications. In case of any
reclassification or change of outstanding securities issuable upon exercise of
the Warrants at any time after the Issuance Date (other than a change in par
value, or from par value to no par value or as a result of a subdivision or
combination to which subsection 2(b) applies), or in case of any consolidation
or merger of the Company with or into any entity or other person (other than a
merger with another entity or other person in which the Company is the surviving
corporation and which does not result in any reclassification or change in the
securities issuable upon exercise of this Warrant Certificate), the holder of
the Warrants shall have, and the Company, or such successor corporation or other
entity, shall covenant in the constituent documents effecting any of the
foregoing transactions that such holder does have the right to obtain, upon the
exercise of the Warrants, in lieu of each share of Common Stock, other
securities, money or other property theretofore issuable upon exercise of a
Warrant, the kind and amount of shares of stock, other securities, money or
other property receivable upon such reclassification, change, consolidation or
merger by a holder of the shares of Common Stock, other securities, money or
other property issuable upon exercise of a Warrant if the Warrants had been
exercised immediately prior to such reclassification, change, consolidation or
merger. The constituent documents effecting such reclassification, change,
consolidation or merger shall provide for adjustments which shall be as nearly
equivalent as may be practicable to the adjustments provided in this subsection
2(a). The provisions of this subsection 2(a) shall similarly apply to successive
reclassifications, changes, consolidations or mergers.

                                    Sch II-3

<PAGE>   24

            Subdivisions and Combinations. If the Company, at any time after the
Issuance Date, shall subdivide its shares of Common Stock into a greater number
of shares, the Exercise price in effect immediately prior to such subdivision
shall be proportionately reduced, and the number of shares of Common Stock
purchasable upon exercise of the Warrants shall be proportionately increased, as
at the effective date of such subdivision, or if the Company shall take a record
of holders of its Common Stock for such purpose, as at such record date,
whichever is earlier. If the Company, at any time after Issuance Date, shall
combine its shares of Common Stock into a smaller number of shares, the Exercise
Price in effect immediately prior to such combination shall be proportionately
increased, and the number of shares of Common Stock purchasable upon exercise of
the Warrants shall be proportionately reduced, as at the effective date of such
combination, or if the Company shall take a record of holders of its Common
Stock for purposes of such combination, as at such record date, whichever is
earlier.

            Dividends and Distributions. If the Company at any time after the
Issuance Date shall declare a dividend on its Common Stock payable in stock or
other securities of the Company to the holders of its Common Stock, the holder
of this Warrant Certificate shall, without additional cost, be entitled to
receive upon any exercise of a Warrant, in addition to the Common Stock to which
such holder would otherwise be entitled upon exercise, the number of shares of
stock or other securities which such holder would have been entitled to receive
if he had been a holder immediately prior to the record date for such dividend
(or, if no record date shall have been established, the payment date for such
dividend) of the number of shares of Common Stock purchasable on exercise of
such Warrant immediately prior to such record date or payment date, as the case
may be.

            Certain Issuances of Securities. If the Company at any time after
the Issuance Date shall issue any additional shares of Common Stock (otherwise
than as provided in paragraphs (a) through (c) of this Section 2) at a price per
share less than the average Price per share of Common Stock for the 20 trading
days immediately preceding the date of the authorization of such issuance (the
"Market Price") by the Board of Directors, then the Exercise Price upon each
such issuance shall be adjusted to that price determined by multiplying the
Exercise Price by a fraction:

     the numerator of which shall be the sum of (1) the number of shares of
Common Stock outstanding immediately prior to the issuance of such additional
shares of Common Stock multiplied by the Market Price, and (2) the
consideration, if any, received by the Company upon the issuance of such
additional shares of Common Stock, and

     the denominator of which shall be the Market Price multiplied by the total
number of shares of Common Stock outstanding immediately after the issuance of
such additional shares of Common Stock.

     No adjustments of the Exercise Price shall be made under this paragraph (d)
upon the issuance of any additional shares of Common Stock that (w) are issued
pursuant to thrift plans, stock purchase plans, stock bonus plans, stock option
plans, employee stock ownership plans, incentive or profit sharing arrangements
for the benefit of employees ("Employee Benefit Plans") that are approved by the
Board of Directors of the Company or its compensation committee and that
otherwise would cause an adjustment under this paragraph (d); provided that the
aggregate number

                                    Sch II-4

<PAGE>   25

of shares of Common Stock so issued (including the shares issued pursuant to any
options, rights or warrants or convertible or exchangeable securities issued
under such Employee Benefit Plans containing the right to purchase shares of
Common stock) pursuant to Employee Benefit Plans after the closing date of the
IPO, as adjusted for any stock splits, stock dividends or subdivisions or
combinations of Common Stock prior to the Expiration Date, shall not in the
aggregate exceed 10% of the Company's outstanding Common Stock at the time of
such issuance; (x) are issued pursuant to any Common Stock Equivalent (as
hereinafter defined) (i) if upon the issuance of any such Common Stock
Equivalent, any such adjustments shall previously have been made pursuant to
paragraph (e) of the Section 2, (ii) if no adjustment was required pursuant to
paragraph (e) of this Section 2, or (iii) if such Common Stock Equivalent was
issued prior to this Warrant Certificate; or (z) are issued pursuant to a public
offering by the Company.

            Common Stock Equivalents. If the Company shall, after the Issuance
Date, issue any security or evidence of indebtedness which is convertible into
or exchangeable for Common Stock or any ("Convertible Security"), other than
pursuant to Employee Benefit Plans (together with Convertible Securities,
"Common Stock Equivalent "), or if, after any such issuance, the price per Share
for which additional shares of Common Stock may be issuable thereunder is
amended, then the Exercise Price upon each such issuance or amendment shall be
adjusted as provided in subsection (d) on the basis that (i) the maximum number
of additional shares of Common Stock issuable pursuant to all such Common Stock
Equivalents shall be deemed to have been issued as of the earlier of (a) the
date on which the Company shall enter into a firm contract for the issuance of
such Common Stock Equivalent, or (b) the date of actual issuance of such Common
Stock Equivalent; and (ii) the aggregate consideration for such maximum number
of additional shares of Common Stock shall be deemed to be the minimum
consideration received and receivable by the Company for the issuance of such
additional shares of Common Stock pursuant to such Common Stock Equivalent;
provided, however, that no adjustment shall be made pursuant to this subsection
(e) unless the consideration received and receivable by the Company per share of
Common Stock for the issuance of such additional shares of Common Stock pursuant
to such Common Stock Equivalent is less than the Market Price. No adjustment of
the Exercise Price shall be made under this subsection (e) upon the issuance of
any Convertible Security which is issued pursuant to the exercise of any
warrants or other subscription or purchase rights therefor, if any adjustment
shall previously have been made in the Exercise Price then in effect upon the
issuance of such warrants or other rights pursuant to this subsection (e).

            Miscellaneous. The following provisions shall be applicable to the
making of adjustments in the Exercise Price hereinbefore provided in this
Section 2:

               The consideration received by the Company shall be deemed to be
the following: (I) to the extent that any additional shares of Common Stock or
any Common Stock Equivalent shall be issued for cash consideration, the
consideration received by the Company therefor, or, if such additional shares of
Common Stock or Common Stock Equivalent are offered by the Company for
subscription, the subscription price, or, if such additional shares of Common
Stock or Common Stock Equivalent are sold to underwriters or dealers for public
offering without a subscription offering, the initial public offering price, in
any such case excluding any amounts paid or receivable for accrued interest or
accrued dividends and without deduction of any compensation,

                                    Sch II-5

<PAGE>   26

discounts, commissions or expenses paid or incurred by the Company for and in
the underwriting of , or otherwise in connection with, the issue thereof; (II)
to the extent that such issuance shall be for a consideration other than cash,
then, except as herein otherwise expressly provided, the fair value of such
consideration at the time of such issuance as determined in good faith by the
Board of Directors, as evidenced by a certified resolution of the Board of
Directors delivered to the holder of this Warrant Certificate setting forth such
determination. The consideration for any additional shares of Common Stock
issuable pursuant to any Common Stock Equivalent shall be the consideration
received by the Company for issuing such Common Stock Equivalent, plus the
additional consideration payable to the Company Upon the exercise, conversion or
exchange of such Common Stock or Common Stock Equivalent in payment of
satisfaction of any dividend upon any class of stock other that Common Stock,
the Company shall be deemed to have received for such additional shares of
Common Stock or Common Stock Equivalent (which shall not be deemed to be a
dividend payable in, or other distribution of, Common Stock under subsection (c)
above) consideration equal to the amount of such dividend so paid or satisfied.

            Upon the expiration of the right to convert, exchange or exercise
any Common Stock Equivalent the issuance of which effected an adjustment in the
Exercise Price, if any such Common Stock Equivalent shall not have been
converted, exercised or exchanged, the number of shares of Common Stock deemed
to be issued and outstanding because they were issuable upon conversion,
exchange or exercise of any such Common Stock Equivalent shall no longer be
computed as set forth above, and the Exercise Price shall forthwith be
readjusted and thereafter be the price which it would have been (but reflecting
any other adjustments in the Exercise Price made pursuant to the provisions of
subsection (d) after the issuance of such Common Stock Equivalent) had the
adjustment of the Exercise Price made upon the issuance or sale of such Common
Stock Equivalent been made on the basis of the issuance only of the number of
additional shares of Common Stock actually issued upon exercise, conversion or
exchange of such Common Stock actually so issued shall be deemed to have been
issued and only the consideration actually received by the Company (computed as
in subparagraph (i) of this paragraph (f) shall be deemed to have been received
by the Company.

            The number of shares of Common Stock at any time outstanding shall
not include any shares thereof then directly or indirectly owned or held by or
for the account of the Company or its wholly owned subsidiaries.

            For the purpose of this Section 2, the term "shares of Common Stock"
shall mean shares of (i) the class of stock designated as the Common Stock at
the date hereof or (ii) any other class of stock resulting from successive
changes or reclassifications of such shares consisting solely of changes in par
value, or from par value to no value, or from no par value to par value. If at
any time, because of an adjustment pursuant to subsection (a), the Warrants
shall entitle the holders to purchase any securities other than shares of Common
Stock, thereafter the number of such other securities so purchasable upon
exercise of each Warrant and the Exercise Price of such securities shall be
subject to adjustment from time to time in a manner and on terms as nearly
equivalent as practicable to the provisions with respect to the Common Stock
contained in this Section 2.

                                    Sch II-6

<PAGE>   27

            Calculation of Exercise Price. The exercise Price in effect from
time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

            Notice of Adjustments. Whenever the Exercise Price or the number of
shares of Common Stock is required to be adjusted as provided in Section 2, the
Company shall forthwith compute the adjusted Exercise Price or the number of
shares of Common Stock issuable and shall prepare and mail to the holder hereof
a certificate setting forth such adjusted Exercise Price or such number of
shares of Common Stock, showing in reasonable detail the facts upon which the
adjustment is based.

            Voluntary Reduction. (a) The Company may at its option, but shall
not be obligated to, at any time during the term of the Warrants, reduce the
then current Exercise Price by any amount selected by the Board of Directors;
provided that is the Company elects so to reduce the then current Exercise Price
to, such reduction shall be irrevocable during its effective period and remain
in effect for a minimum of 30 days following the date of such election, after
which time the Company may, at its option, reinstate the Exercise Price in
effect prior to such reduction. Whenever the Exercise Price is reduced, the
Company shall mail to the holder a notice of the reduction at least 30 days
before the date the reduced Exercise Price takes effect, stating the reduced
Exercise Price and the period for which such Exercise Price will be in effect.

            The Company may make such decreases in the Exercise Price, in
addition to those required or allowed by this Section 4, as shall be determined
by it, as evidenced by a certified resolution of the Board of Directors
delivered to the holders, to be advisable to avoid or diminish any income tax to
the holder resulting from any dividend or distribution of stock or issuance of
rights or warrants to purchase or subscribe for stock or from any event treated
as such for income tax purposes.

     Notices to Warrant Holders. In the event:

            (a) the Company shall authorize any consideration or merger to which
the Company is a party and for which approval of any stockholders of the Company
is required, or of the conveyance or sale of all or substantially all of the
assets of the Company, or of any reclassification or change if the Common Stock
or other securities issuable upon exercise of the Warrants (other than a change
in par value, or from par value to no par value, or from no par value to par
value or as result of a subdivision or combination), or a tender offer or
exchange offer for shares of Common Stock (or other securities issuable upon the
exercise of the Warrants); or

            (b) the Company shall declare any dividend (or any other
distribution) on the Common Stock or any other class of its capital stock ; or

            (c) the Company shall authorize the granting to the holders of
Common Stock or any other class of its capital stock of rights or warrants to
subscribe for or purchase any shares of any class or series of capital stock or
any other securities convertible into or exchangeable for shares of stock; or

                                    Sch II-7

<PAGE>   28

            (d) of the voluntary or involuntary dissolution, liquidation or
winding up of the Company; then the Company shall cause to be sent to the holder
hereof, at least 30 days prior to the applicable record date hereinafter
specified, or promptly in the case of events for which there is no record date,
a written notice stating (x) the date for the determination of the holders of
record of shares of Common Stock (or other securities issuable upon the exercise
of the Warrants) entitled to receive any such dividends other distribution, (y)
the initial expiration date set forth in any tender offer or exchange offer for
shares of Common Stock (or other securities issuable upon the exercise of the
Warrants), or (z) the date on which any of the events specified in subsections
(a)-(d) is expected to become effective or consummated, and the date as of which
it is expected that holders of record of shares of Common Stock (or securities
issuable upon the exercise of the Warrants) shall be entitled to exchange such
shares for securities or other property, if any, deliverable upon any such
event. Failure to give such notice or any defect therein shall not affect the
legality or validity of any such event, or the vote upon any such action.

     6. Reports to Warrant Holders. The Company will cause to be delivered, by
first-class mail, postage prepaid, to holder at such holder's address appearing
hereon, or such other address as the holder shall specify, a copy of any reports
delivered by the Company to the holders of Common Stock.

     7. Covenants of the Company. The Company covenants and agrees that:

        (a) Until the Expiration Date, the Company shall at all times reserve
and keep available, out of the aggregate of its authorized but unissued Common
Stock (and other securities), for the purpose for enabling it to satisfy any
obligation to issue shares of Common Stock (and other securities) upon the
exercise of the Warrants, the number of shares of Common Stock (and other
securities) issuable upon the exercise of such Warrants.

        (b) The Company shall pay all expenses, taxes and other charges payable
in connection with the preparation, issuance and delivery of new warrant
certificates on transfer of the Warrants.

        (c) All Common Stock (and other securities) which may be issued upon
exercise of the Warrants shall upon issuance be validly issued, fully paid,
non-assessable and free from all preemptive rights and all taxes, liens and
charges with respect to the issuance thereof, and will not be subject to any
restrictions on voting or transfer thereof except as set forth in any
stockholders agreement.

        (d) All original issue taxes payable in respect of the issuance of
shares of Common Stock to the registered holder hereof upon the exercise of the
Warrants shall be borne by the Company; provided, that the Company shall not be
required to pay any tax or charge imposed in connection with any transfer
involved in the issuance of any certificates representing shares of Common Stock
(and other securities) in any name other than that of the registered holder
hereof, and in such case the Company shall not be required to issue or deliver
any certificate representing shares of Common Stock (and other securities) until
such tax or other charge has been paid or it has been established to the
Company's satisfaction that no such tax or charge is due.

                                    Sch II-8

<PAGE>   29

            (e) As soon as practicable after the receipt from the holder of this
Warrant Certificate of notice of the exercise of a number of warrants sufficient
to require a filing under the Hart-Scott- Rodino Antitrust Improvements Act of
1976 and the rules, regulations and formal interpretations thereunder, as
amended from time to time (the "HSR Act:), but in any event no later than the
10th business day after receipt of such notice, the Company will (i) prepare and
file with Antitrust Division of the Department of Justice (the "DOJ") and the
Federal Trade Commission (the "FTC") the Notification and Report Form
(accompanied by all documentary attachments contemplated thereby) required by
the HSR Act, (ii) upon request of the holder, request early termination of the
waiting period imposed by the HSR Act, (iii) coordinate and cooperate with the
holder in responding to formal and informal requests for additional information
and documentary material from the DOJ and the FTC in connection with such
filing, (iv) use its best efforts to take, or cause to be taken, all reasonable
action to do, or cause to be done, all things necessary and appropriate to
permit the issuance to the holder of the shares of Common Stock issuable upon
the exercise of the warrants with respect to which any filing is required under
the HSR Act, and (v) reimburse the holder for the entire amount of any filing
fee or any other costs and expenses incurred by the holder in connection
therewith (including legal fees), or as required to be paid under the HSR Act.

        8. No Rights as Stockholder. The holder of the Warrants shall not, by
virtue of holding such Warrants, be entitled to any rights of a stockholder of
the Company either at law or in equity, and the rights of the holder of the
Warrants are limited to those expressed herein.

        9. Notices. All notices provided for hereunder shall be in writing and
may be given by registered or certified mail, return receipt requested, telex,
telegram, telecopier, air courier guaranteeing overnight delivery of personal
delivery, if to the holder at the following address:

               ------------------------------
               ------------------------------
               ------------------------------
               ------------------------------
               ------------------------------

           and, if to the Company:

               Brigham Exploration Company
               6300 Bridge Point Parkway
               Building 2, Suite 500
               Austin, Texas  78730
               Attention: Chief Financial Officer
               Telecopier: (512) 472-3400

        10. Governing Law. This Warrant Certificate shall be governed by and
construed in accordance with the laws of the State of Texas without principles
to conflict of laws.

                                    Sch II-9

<PAGE>   30

        11. Lost, Stolen, Mutilated or Destroyed Warrant Certificates. Upon
receipt by the Company of evidence reasonably satisfactory to it of the
ownership of and the loss, theft, destruction or mutilation of any Warrant
Certificate, then, in the absence of notice to the Company that such warrant
Certificate has been acquired by a bona fide purchaser, the Company shall
execute and deliver, in exchange for or in lieu of the lost, stolen, destroyed
or mutilated Warrant Certificate, a substitute Warrant Certificate of the same
tenor and evidencing a like number of Warrants.

     IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to be
executed as of February _____, 2000, by the undersigned, thereunto duly
authorized.

                                       BRIGHAM EXPLORATION COMPANY

                                       By:
                                          --------------------------------------
                                                Karen E. Lynch
                                                Vice President

                                    Sch II-10

<PAGE>   31

                                    EXHIBIT A

                              ELECTION TO EXERCISE

          [To be executed on exercise of the Warrant evidenced by this
                 Warrant Certificate pursuant to Section 1(a)]

          TO:    Brigham Exploration Company

                  The undersigned, the holder of the Warrants evidenced by the
attached Warrant Certificate, hereby irrevocably elects to exercise Warrants,
and herewith makes payment of_____________________ delivered to
_____________________________, whose address is _______________________________.

                  The undersigned hereby directs payment of the Exercise Price
to be made by application of [specify Notes/Warrant Shares/cash].

          Dated:
                -------------------

                                      Name of Registered Holder:
                                                                ----------------
                                      Signature:
                                                --------------------------------
                                      Title:
                                            ------------------------------------
                                      Address:
                                              ----------------------------------

         Notice: The above signature(s) must correspond with the name as written
on the face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                    Sch II-11

<PAGE>   32

                                   EXHIBIT A-1

                              ELECTION TO EXERCISE

          [To be executed on exercise of the Warrant evidenced by this
Warrant Certificate pursuant to Section 1(c)]

          TO:    Brigham Exploration Company

                  The undersigned, the holder of the Warrants evidenced by the
         attached Warrant Certificate, hereby elects to exercise _____________
         Warrants.

[ ]      The undersigned elects to receive shares of Common Stock of the Company
         in an amount equal to the Appreciation Amount (as defined in the
         Warrant), in accordance with Section 1(c) of the Warrant, and requests
         that the securities issuable hereunder be issued in the name of
         _________________ whose address is ________________________________.

         Dated:
               --------------------

                                      Name of Registered Holder:
                                                                ----------------
                                      Signature:
                                                --------------------------------
                                      Title:
                                            ------------------------------------
                                      Address:
                                              ----------------------------------

         Notice: The above signature(s) must correspond with the name as written
on the face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                    Sch II-12

<PAGE>   33

                                  SCHEDULE III*

<TABLE>
<CAPTION>

SHARES OF                             STOCK OPTIONS OUTSTANDING
COMMON                 ----------------------------------------------------        WARRANTS **
STOCK OUTSTANDING         VESTED             UNVESTED             TOTAL            OUTSTANDING
-----------------      ------------        ------------        ------------        ------------
<S>                    <C>                 <C>                 <C>                 <C>
14,517,786                  360,671           1,485,390           1,846,061           2,000,000
</TABLE>

*As of February 16, 2000.

**    1. These outstanding warrants do not include up to 5,416,667 warrants that
      may be issued, under certain circumstances, to Shell Capital, Inc. under
      that certain Equity Conversion Agreement dated as of February 17, 2000,
      between the Company and Shell Capital, Inc.

**    2. Of the 2,000,000 warrants outstanding as reflected in the table above,
      the exercise price of 1,000,000 warrants was repriced on February 17,
      2000, from $2.25 per share to $2.02 per share, and the exercise price of
      the remaining 1,000,000 warrants will be repriced from $3.50 per share to
      the average closing market price of Brigham Common Stock for the 20
      trading days subsequent to February 22, 2000.

                                    Sch III-1

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