<PAGE>   1
                                                                    EXHIBIT 4.3






                              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 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 the Initial
Issuance Date, ___% of the outstanding Common Stock on a fully diluted basis
using the treasury stock method.  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 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 $_______.

         "Expiration Date" means the date on which the seventh (7th)
anniversary of the Funding Date occurs.

         "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 of 1933, as amended, and the rules and regulations
promulgated thereunder.

          "Issuance Date" means ______________, 1998.
<PAGE>   2
         "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 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.

         "Securities Purchase Agreement" means the Securities Purchase
Agreement, dated as of ______________, 1998, among the Company, ________ and 
individually and as agent.

         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, 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
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)      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-
<PAGE>   3
         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 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





                                      -3-
<PAGE>   4
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

                          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





                                      -4-
<PAGE>   5
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).

                 (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.





                                      -5-
<PAGE>   6
                          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 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.





                                      -6-
<PAGE>   7
         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.

         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





                                      -7-
<PAGE>   8
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.

         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





                                      -8-
<PAGE>   9
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 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:


         and, if to the Company:

                 Brigham Exploration Company
                 6300 Bride Point Parkway
                 Building 2, Suite 500
                 Austin, Texas 78730
                 Attention: Craig M. Fleming
                 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.





                                      -9-
<PAGE>   10
         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.


         IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to
be executed this ____ day of __________, 1998 by the undersigned, thereunto
duly authorized.

                                        BRIGHAM EXPLORATION COMPANY


                                        By:
                                           ------------------------------
                                           Craig M. Fleming 
                                           Chief Financial Officer





                                      -10-
<PAGE>   11
                                   EXHIBIT A

                              ELECTION TO EXERCISE
          [To be executed on exercise of the Warrants evidenced by
                          this Warrant Certificate]

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 ___________________________________________________
($_______________) 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 ___________________________________________.

         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.





                                      -11-
<PAGE>   12
                                   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
______________________________________________________  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.





                                      -12-