<PAGE>

                                                                   Exhibit 10.73

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 THIS WARRANT CERTIFICATE AND IN A SECURITIES PURCHASE
AGREEMENT, DATED AS OF MARCH 5, 2001, BETWEEN BRIGHAM EXPLORATION COMPANY AND
THE INITIAL HOLDER 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.

                               WARRANT CERTIFICATE

Number of Warrants: 371789                                       Warrant No. 2
                   -------                                                  ----

      This Warrant certificate ("Warrant Certificate") certifies that, for value
received, DLJ ESC II, L.P. 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. 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 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 day
      immediately following the date that Stockholder Approval, as defined in
      the Securities Purchase Agreement, is obtained 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 $4.75 per share.

            "Expiration Date" means the tenth anniversary of the Closing Date.

            "Issuance Date" means the Closing Date.
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            "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.

            "Preferred Stock" means shares of the Series A Preferred Stock, par
      value $0.01 per share, of the Company.

            "Preferred Value" per share of Preferred Stock means the Stated
      Value of such Share, plus, without duplication, all accrued and unpaid
      dividends on such share to and including the applicable date of Warrant
      exercise.

            "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 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 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.
      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.

            "Securities Purchase Agreement" means the Securities Purchase
      Agreement, dated as of March 5, 2001, between the Company and the
      Investors.

            "Stated Value" means the stated value per share of Preferred Stock,
      which is $20.00 per share.

      Section 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 10 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, for the number of Warrants being exercised by either: (1) bank
draft or cashiers check, or (2) provided that the Company receives at least 5
days prior notice and subject to Section 1(d), delivery to the Company of
certificate(s) representing a number of shares of Preferred Stock having an
aggregate Preferred Value equal to the aggregate Exercise Price for the number
of Warrants being exercised. If the aggregate Preferred Value of the Preferred
Stock delivered in payment of the aggregate Exercise Price exceeds (because of
fractional shares) the aggregate Exercise Price for the number of Warrants being
exercised; then (subject to Section 1(d)) the Company will promptly pay to the
holder of the Warrants in cash such excess amount; provided that such excess
amount shall in no event be more than the Preferred Value of one share of
Preferred


                                      -2-
<PAGE>

Stock. 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 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 10 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 shall expire and the rights of the
holder shall become void and of no effect.

            (c) Upon surrender of this Warrant Certificate in conformity with
the foregoing provisions, the Company shall transfer to the holder of this
Warrant Certificate appropriate evidence of ownership of the shares of Common
Stock or other securities or property (including any money) to which the holder
is entitled, registered or otherwise placed in, or payable to the order of, the
name or names of the holder or such transferee as may be directed in writing by
the holder, and shall deliver such evidence of ownership and any other
securities or property (including any money) to the Person or Persons entitled
to receive the same, together with an amount in cash in lieu of any fraction of
a share.

            (d) In connection with payment of the Exercise Price with shares of
Preferred Stock, the Company may require that at the time of such exercise it
receive representations and warranties from the applicable holder of the
Warrants regarding such holder's title to the Preferred Stock and the lack of
encumbrances thereon. If the Company is unable to consummate an exercise of
Warrants through payment of the Exercise Price with shares of Preferred Stock
because of any limitations contained or construed in the Delaware General
Corporation Law, the Company shall use its best efforts to take all such action
as may be necessary to place the Company in a position to do so. In the event
the Company, after the taking of any action by it as contemplated above, is
unable to consummate such exercise, the Company shall accept such number of
shares of Preferred Stock in payment as it shall then be authorized to do so
under the Delaware General Corporation Law.

            (e) The Company shall not be required to issue a fractional share of
Common Stock upon the exercise of Warrants. As to any fraction of a share which
the Warrant holder would otherwise be entitled to purchase upon such exercise,
the Company may pay a cash adjustment in respect of such fraction in an amount
equal to the same fraction of the Price per share of Common Stock on the date of
exercise.

      Section 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:


                                      -3-
<PAGE>

            (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 Section 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 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 Section 2(a). The provisions of this Section 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


                                      -4-
<PAGE>

the case may be.

            (d) Certain Issuances of Securities. Subject to Section 2(f), if the
Company at any time after the Issuance Date shall issue any additional shares of
Common Stock (otherwise than as provided in subsections (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 or
its compensation committee (as applicable), 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

            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 Section 2(d) upon
the issuance of any additional shares of Common Stock that (v) are issued
pursuant to any grant or award made prior to the Issuance Date under any thrift
plan, stock purchase plan, stock bonus plan, stock option plan, employee stock
ownership plan, incentive or profit sharing arrangement or other benefit or
compensation plan for the benefit of the Company's officers, directors and/or
employees ("Employee Benefit Plans") that has been approved by the Board of
Directors of the Company or its compensation committee and that otherwise would
cause an adjustment under this Section 2(d); (w) are issued pursuant to any
grant or award made on or after the Issuance Date under any Employee Benefit
Plan if the "Market Price" of any such issuance is not less than the lesser of
the Market Price as determined above and the "Fair Market Value", as defined
under the applicable Employee Benefit Plan, on the date of Board or compensation
committee authorization; (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
Section 2(e), (ii) if no adjustment was required pursuant to Section 2(e), or
(iii) if such Common Stock Equivalent was issued prior to this Warrant
Certificate; (y) are issued pursuant to a public offering by the Company; or (z)
results in an adjustment pursuant to Section 2(f).

            (e) Common Stock Equivalents.

            i. Subject to Section 2(f), 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"), then
      the Exercise Price upon each such issuance shall be adjusted as provided
      in Section 2(d) on the


                                      -5-
<PAGE>

      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 date of 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.

            ii. Notwithstanding the foregoing, no adjustment shall be made
      pursuant to this Section 2(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 Section 2(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
      Section 2(e). No adjustment shall be made under this Section 2(e) if an
      adjustment is to be made under Section 2(f). No adjustment shall be made
      as a result of adjustment in the exercise or conversion price of Common
      Stock Equivalents, if those adjustments occur by the terms of such Common
      Stock Equivalents.

            (f) Special Adjustments of Exercise Price. Notwithstanding anything
to the contrary in Section 2(d) or Section 2(e), this Section 2(f) shall govern
adjustments to the Exercise Price for the transactions described in this Section
2(f).

            i. If the Company at any time after the Issuance Date and prior to
      the second anniversary of the Issuance Date shall issue any additional
      shares of Common Stock (otherwise than as provided in subsections (a)
      through (c) of Section 2; pursuant to any Employee Benefit Plan; pursuant
      to any Common Stock Equivalent outstanding as of the Issuance Date) or
      upon the issuance of any such Common Stock, any adjustments shall
      previously have been made pursuant to Section 2(e) or Section 2(f)(ii);
      and the New Stock Issue Price (defined below) of such additional shares is
      less than the Exercise Price then in effect, then the Exercise Price upon
      each such issuance shall be adjusted to the New Stock Issue Price of such
      additional shares. The "New Stock Issuance Price" shall be determined by
      dividing the total amount of consideration received by the Company for
      such issue or sale by the number of shares of Common Stock issued or sold.

            ii. If the Company at any time after the Issuance Date and prior to
      the second anniversary of the Issuance Date, issues any Common Stock
      Equivalent (which by definition excludes Employee Benefit Plan securities)
      (otherwise than as provided in subsections (a) through (c) of Section 2;
      or pursuant to any Common Stock Equivalent outstanding as of the Issuance
      Date) and the New CSE Exercise Price (defined below) of such Common Stock
      Equivalents is less than the Exercise Price then in effect, then the
      Exercise Price upon each such issuance shall be adjusted to the New CSE
      Exercise Price of such Common Stock Equivalents. The "New CSE Exercise
      Price" shall be determined by dividing (x) the total


                                      -6-
<PAGE>

      amount, if any, received or receivable by the Company as consideration for
      the issuance of such Common Stock Equivalents, plus the minimum aggregate
      amount of additional consideration payable to the Company upon the
      exercise, conversion or exchange of such Common Stock Equivalents, plus,
      in the case of any such Common Stock Equivalents which relate to
      Convertible Securities, the minimum aggregate amount of additional
      consideration, if any, payable to the Company upon the conversion or
      exchange of such Convertible Securities, by (y) the total maximum number
      of shares of Common Stock issuable upon the exercise, conversion or
      exchange of all such Common Stock Equivalents.

      (g) 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 Section
      2(c) above) consideration equal to the amount of such dividend so paid or
      satisfied. In the event additional shares of Common Stock or Common Stock
      Equivalents are issued together with other shares or securities or other
      assets of the Company or its subsidiaries for consideration which covers
      both, the consideration for such shares of Common Stock and Common Stock
      Equivalents shall be computed based on the respective portions of such
      consideration so received, computed as provided in this Section 2(g)i., as
      determined and allocated in good faith by the Board of Directors of the
      Company.


                                      -7-
<PAGE>

            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 Section 2(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 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 this Section 2(f)(i)) 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. Upon each adjustment of the Exercise Price as a result of the
      calculations made in Section 2(d), (e) and (f) hereof, this Warrant shall
      thereafter evidence the right to purchase, at the adjusted Exercise Price,
      that number of shares of Common Stock obtained by (i) multiplying the
      number of shares covered by this Warrant immediately prior to such
      adjustment of the number of shares by the Exercise Price in effect
      immediately prior to such adjustment of the Exercise Price and (ii)
      dividing the product so obtained by the Exercise Price in effect
      immediately after such adjustment of the Exercise Price.

            v. 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 par value, or from
      no par value to par value. If at any time, because of an adjustment
      pursuant to Section 2(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.

            (h) 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.


                                      -8-
<PAGE>

      Section 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.

      Section 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.

            (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.

      Section 5. Mandatory Exercise.

            (a) If (i) the Price of the Common Stock is greater than 150% of the
Exercise Price (as adjusted to reflect any stock split, combination,
reclassification, recapitalization, exchange, stock dividend or other
distribution payable in Common Stock with respect to shares of Common Stock) for
sixty (60) consecutive trading days in the principal market in which the Common
Stock is traded and (ii) the Company gives written notice (the "Company Notice")
to the holder hereof of the satisfaction of the condition in clause (i), then
within fifteen (15) days after the effective date of the Company Notice, the
holder hereof shall exercise all of the Warrants. If required by this Section 5,
the holder hereof agrees to exercise the Warrants, and to purchase shares of
Common Stock pursuant to the terms of this Warrant Certificate. If the holder
has not fulfilled its obligations to exercise the Warrants pursuant to this
Section 5 within fifteen (15) days after the holder's receipt of the Company
Notice, then (without limiting the Company's available remedies) (A) the
obligations of holder under this Section 5 shall continue but the purchase
rights otherwise represented by this Warrant Certificate shall terminate, (B)
the Company may thereafter refuse, in its sole discretion, to allow holder to
exercise the Warrants (including pursuant to this Section 5), (C) all
obligations of the Company under Sections 3, 6, 7 and 8 shall terminate, (D) no
further adjustments to the Exercise Price shall be made unless the Company in
its sole discretion consents in writing. Each Warrant holder's obligations under
this Section 5(a) shall be subject to the expiration or termination of all
waiting periods (and any extensions thereof) applicable to exercise of such
holder's Warrants


                                      -9-
<PAGE>

under the HSR Act (as defined below); provided that such holder shall have
certified in writing to the Company that a filing under the HSR Act is required
and provided further that such holder shall use its best efforts to cause the
expiration or termination of such waiting period to occur as promptly as
practicable.

            (b) Holder represents and warrants to the Company that holder has
full corporate power and authority to execute, deliver, and perform this Warrant
Certificate and to consummate the transactions contemplated hereby. The
execution, delivery, and performance by holder of this Warrant Certificate have
been duly authorized by all necessary corporate action of holder. This Warrant
Certificate has been duly executed and delivered by holder and constitutes a
valid and legally binding obligation of holder, enforceable against holder in
accordance with its terms, except that such enforceability may be limited by (i)
applicable bankruptcy, insolvency, reorganization, moratorium, and similar laws
affecting creditors' rights generally and (ii) general equitable principles
(regardless of whether such enforceability is considered in a proceeding in
equity or at law).

            (c) The right to require exercise of the Warrants is hereby declared
by the parties hereto to be a unique right, the loss of which is not readily
susceptible to monetary quantification. Consequently, the parties hereto agree
that an action for specific performance of the exercise and purchase obligations
created by this Section 5 is an available remedy for the breach of the
provisions of this Section 5. If the Company is forced to institute legal
proceedings to enforce its rights in accordance with the provisions of this
Section 5, it shall be entitled to recover its reasonable attorneys' fees and
court costs incurred in enforcing such rights.

            (d) Holder is executing this Warrant Certificate in order to make
and agree to the covenants, representations and warranties of holder contained
in this Section 5, which shall be binding upon the holder's successors and
assigns.

      Section 6. 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 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, other
than dividends on the Shares, as defined in the Securities Purchase Agreement;
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


                                      -10-
<PAGE>

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.

      Section 7. 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.

      Section 8. 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 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 the Securities
Purchase Agreement, any stockholders agreement and except for restrictions
arising under state or federal securities laws.

            (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


                                      -11-
<PAGE>

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.

            (e) As soon as practicable after the receipt from the holder of this
Warrant Certificate of notice of the intent to 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") (and after the receipt,
if applicable, of the notice referred to in Rule 803.5 of the HSR Act), but in
any event no later than the 15th business day after receipt of such notice(s),
the Company will (i) if required by the HSR Act, 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, and (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.
Notwithstanding the foregoing, if the holder is required to file with the DOJ
and FTC the Notification and Report Form solely as a result of its holding
and/or purchasing shares of Common Stock issued pursuant to this Warrant (with
no regard to any other securities held by such holder or its affiliates) and the
holder certifies such fact to the Company in writing, the Company agrees to
promptly reimburse the holder for all fees and expenses for the preparation and
filing of such form, including all legal expenses and filing fees.

      (f) The Company will not change the par value of the Common Stock from par
value $0.01 per share to any higher par value which exceeds the Exercise Price
then in effect, and will reduce the par value of the Common Stock upon any event
described in Section 2 that would, but for this provision, reduce the Exercise
Price below the par value of the Common Stock.

      Section 9. 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.

      Section 10. 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) sent by prepaid overnight courier service, or
(iii) 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 the holder, to such holder at:


                                      -12-
<PAGE>

                  277 Park Avenue
                  New York, New York 10172
                  Attention: Ivy Dodes/Nicole Arnaboldi
                  Telefax: 212-892-7272

            with a copy to:

                  Gardere Wynne Sewell, LLP
                  1000 Louisiana, Suite 3400
                  Houston, Texas 77002
                  Attention: N.L. Stevens III
                  Telefax: 713-276-5807

            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

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, or (ii) if sent by telecopy or facsimile transmission, when
the answer back is received.

      Section 11. Governing Law. This Warrant Certificate shall be governed by
and construed in accordance with the laws of the State of New York without
regard to principles of conflict of laws.

      Section 12. 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.

      Section 13. Transfer. Subject to Section 14 and the Securities Purchase
Agreement, transfer of Warrants, in whole or in part, shall be registered on the
books of the Company to be maintained for such purposes, upon surrender of the
Warrant Certificate representing such Warrants at the principal office of the
Company referred to in Section 10, together with a written assignment
substantially in the form of Exhibit B to this Warrant Certificate and a written
agreement, in form reasonably satisfactory to the Company, setting forth the new
Warrant holder's agreement to be bound by all of the terms of this Warrant
Certificate (including without limitation Section 14) and


                                      -13-
<PAGE>

Section 5.5 of the Securities Purchase Agreement, each duly executed by the
holder, and funds sufficient to pay any transfer taxes payable by such holder
upon the making of such transfer. Upon such surrender and, if required, such
payment, the Company shall execute and deliver a new Warrant Certificate or
Warrant Certificates in the name of the assignee or assignees and in the
denomination specified in such instrument of assignment, and shall issue to the
assignor a new Warrant Certificate or Warrant Certificates evidencing the
portion of the old Warrant Certificate not so assigned, and the old Warrant
Certificate shall promptly be canceled.

      Section 14. Restrictions on Transferability. The Warrant Certificate
represents Warrants referred to in the Securities Purchase Agreement. Said
Securities Purchase Agreement is hereby incorporated by reference in and made a
part of this instrument and is hereby referred to for a description of certain
limitations of rights, obligations, duties and immunities thereunder of the
Company and the holders, and in the event of any conflict between the terms of
this Warrant Certificate and the provisions of the Securities Purchase
Agreement, the provisions of the Securities Purchase Agreement shall control.

            IN WITNESS WHEREOF, the Company has caused this Warrant Certificate
to be executed as of March 5, 2001, by the undersigned, thereunto duly
authorized.

                                        BRIGHAM EXPLORATION COMPANY

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


                                        DLJ ESC II, LP

                                        DLJ LBO Plans Management Corporation,
                                        General Partner

                                        By: /s/ Steven A. Webster
                                           -------------------------------------
                                           Name:  Steven A. Webster
                                           Title: Attorney-in-Fact


                                      -14-