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                                                                  EXHIBIT 10.13


                               WARRANT CERTIFICATE


Number of Warrants: 750,000                                 Warrant No.  A-6

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

         JOINT ENERGY DEVELOPMENT INVESTMENTS II LIMITED PARTNERSHIP

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, 3.1015% 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-4 issued to Joint Energy Development Investments II
Limited Partnership as of March 26, 1999, which in turn had amended and restated
Warrant No. A-2 issued to Joint Energy Development Investments II Limited
Partnership 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




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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 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



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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 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





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     number of additional shares of Common Stock actually issued upon exercise,
     conversion or exchange of such Common Stock Equivalent and thereupon only
     the 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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<PAGE>   11


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:

        Joint Energy Development Investments II Limited Partnership
        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-