<PAGE>   1
                                                                    EXHIBIT 10.4


                                                                  Execution Copy

THIS WARRANT AGREEMENT AND THE SECURITIES ISSUABLE UPON EXERCISE OF THE WARRANTS
HEREIN HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR
THE SECURITIES LAWS OF ANY STATE AND MAY NOT BE TRANSFERRED IN VIOLATION OF SUCH
ACT, THE RULES AND REGULATIONS PROMULGATED THEREUNDER, ANY SUCH STATE SECURITIES
LAWS OR THE PROVISIONS OF THIS WARRANT AGREEMENT.



                                WARRANT AGREEMENT


                                     for the


                            Purchase of Common Stock

                                 By and Between



                           BRIGHAM EXPLORATION COMPANY

                                       and

                                BANK OF MONTREAL



                                   Dated as of

                                  July 19, 1999

<PAGE>   2
                                TABLE OF CONTENTS

<TABLE>
<CAPTION>
                                                                                                               Page

<S>                                                                                                            <C>
1.       DEFINITIONS.............................................................................................2

2.       ISSUANCE AND EXERCISE OF WARRANTS.......................................................................6

         2.1      Issuance of Warrants...........................................................................6

         2.2      Manner of Exercise.............................................................................6

         2.3      Payment of Taxes...............................................................................8

         2.4      Fractional Shares..............................................................................8

         2.5      Continued Validity.............................................................................9

3.       TRANSFERS, DIVISION AND COMBINATION.....................................................................9

         3.1      Transfer.......................................................................................9

         3.2      Division and Combination.......................................................................9

         3.3      Expenses.......................................................................................9

         3.4      Maintenance of Books...........................................................................9

4.       ADJUSTMENTS.............................................................................................9

         4.1      Stock Dividends, Subdivisions and Combinations.................................................9

         4.2      Certain Other Distributions...................................................................10

         4.3      Issuance of Additional Shares of Stock........................................................11

         4.4      Issuance of Warrants or Other Rights..........................................................11

         4.5      Issuance of Convertible Securities............................................................12

         4.6      Superseding Adjustment........................................................................13

         4.7      Other Provisions Applicable to Adjustments Under This Section.................................13

         4.8      Reorganization, Reclassification, Merger, Consolidation or Disposition of Assets..............14

5.       NOTICES TO WARRANT HOLDERS.............................................................................15

         5.1      Notice of Adjustments.........................................................................15

         5.2      Notice of Certain Corporate Action............................................................15

6.       REPRESENTATIONS AND WARRANTIES.........................................................................16

7.       CERTAIN COVENANTS......................................................................................17

         7.1      No Impairment.................................................................................17

         7.2      Reservation and Authorization of Common Stock; Registration with, or Approval of, any
                  Governmental Authority........................................................................18

8.       TAKING OF RECORD; STOCK AND WARRANT TRANSFER BOOKS.....................................................18
</TABLE>


                                       i
<PAGE>   3
<TABLE>
<S>                                                                                                             <C>
9.       RESTRICTIONS ON TRANSFERABILITY........................................................................18

         9.1      Restrictive Legend............................................................................18

         9.2      Notice of Proposed Transfers; Requests for Registration.......................................19

         9.3      Incidental Registration.......................................................................19

         9.4      Registration Procedures.......................................................................20

         9.5      Expenses......................................................................................21

         9.6      Indemnification and Contribution..............................................................21

         9.7      Termination of Restrictions...................................................................24

         9.8      Listing on Securities Exchange................................................................25

10.      SUPPLYING INFORMATION..................................................................................25

11.      LOSS OR MUTILATION.....................................................................................25

12.      OFFICE OF THE ISSUER...................................................................................25

13.      APPRAISAL..............................................................................................25

14.      LIMITATION OF LIABILITY; NO RIGHTS AS STOCKHOLDER......................................................26

15.      MISCELLANEOUS..........................................................................................26

         15.1     Non-waiver and Expenses.......................................................................26

         15.2     Notice Generally..............................................................................26

         15.3     Indemnification...............................................................................27

         15.4     Remedies......................................................................................27

         15.5     Successors and Assigns........................................................................28

         15.6     Complete Agreement; Amendment.................................................................28

         15.7     Severability..................................................................................28

         15.8     Headings......................................................................................28

         15.9     Governing Law; Consent to Jurisdiction and Venue..............................................28

         15.10    Consent to Jurisdiction and Venue.............................................................28

         15.11    Counterparts..................................................................................29

EXHIBITS:

Exhibit A  -Form of Warrant Certificate...................................................................Exh. A-1

Exhibit B  -Form of Cashless Conversion Notice............................................................Exh. B-1

SCHEDULES:

Schedule A -Capital Stock of the Issuer, Including Shares Subject to
           Outstanding Warrants, Options, Conversion Rights, Etc..........................................Sch. A-1
</TABLE>


                                       ii
<PAGE>   4

THIS WARRANT AGREEMENT AND THE SECURITIES ISSUABLE UPON EXERCISE OF THE WARRANTS
HEREIN HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR
THE SECURITIES LAWS OF ANY STATE AND MAY NOT BE TRANSFERRED IN VIOLATION OF SUCH
ACT, THE RULES AND REGULATIONS PROMULGATED THEREUNDER, ANY SUCH STATE SECURITIES
LAWS OR THE PROVISIONS OF THIS WARRANT AGREEMENT.

                                WARRANT AGREEMENT


THIS WARRANT AGREEMENT, dated as of July 19, 1999 (this "Agreement"), is entered
into by and between Brigham Exploration Company, a Delaware corporation
("Issuer"), and Bank of Montreal, a Canadian chartered bank (the "Warrant
Holder" or the "Bank of Montreal").

                              W I T N E S S E T H:


         WHEREAS, Brigham Oil & Gas, L.P., a limited partnership formed under
the laws of the State of Delaware (the "Borrower"), the financial institutions
party to the Credit Agreement referred to below (each a "Lender" and
collectively, the "Lenders"), and the Bank of Montreal, as agent for Lenders
under the Credit Agreement (in such capacity, the "Agent"), are parties to that
certain Credit Agreement, dated as of January 26, 1998, as amended by that
certain First Amendment to Credit Agreement, dated as of August 20, 1998, and
that certain Second Amendment to Credit Agreement, dated as of March 26, 1999
(as so amended, the "Credit Agreement"); and

         WHEREAS, the Borrower advised the Lenders and the Agent that it desired
to amend certain provisions of the Credit Agreement, and the Borrower requested
that the Lenders and the Agent agree to various amendments to certain provisions
of the Credit Agreement; and

         WHEREAS, the Lenders and the Agent have agreed to so amend certain
provisions of the Credit Agreement upon the terms and subject to the conditions
and limitations of the Third Amendment to the Credit Agreement dated as of July
19, 1999 (the "Third Amendment");

         WHEREAS, the Issuer has guaranteed the obligations of the Borrower to
the Lenders and the Agent;

         WHEREAS, as a consequence of the contractual relationships between the
Borrower and the Lenders, the Issuer has and will continue to receive
substantial benefits from the Lenders;

         WHEREAS, in order to induce the Lenders to enter into the Third
Amendment, the Issuer has agreed to execute and deliver this Agreement and to
issue to Bank of Montreal the warrants herein described;



                                       1
<PAGE>   5

         NOW, THEREFORE, in consideration of the premises and mutual covenants
contained herein, the receipt and sufficiency of which is hereby acknowledged,
the parties hereby stipulate and agree as follows:

1.       DEFINITIONS

         As used in this Agreement, the following terms have the respective
         meanings set forth below:

         "Additional Shares of Common Stock" shall mean all shares of Common
         Stock issued by the Issuer after the Closing Date, other than Warrant
         Stock.

         "Affiliate" shall mean as to any Person, any other Person which,
         directly or indirectly, is in control of, is controlled by, or is under
         common control with, such Person. For purposes of this definition,
         "control" (including the terms "controlled by" and "under common
         control with"), as used with respect to any Person, means the power to
         direct or cause the direction of the management and policies of such
         Person, directly or indirectly, whether through the ownership of voting
         securities or by contract or otherwise; and the terms "controlled" and
         "controlling" have the meanings correlative to the foregoing.

         "Appraised Value" shall mean, in respect of any share of Common Stock
         on any date herein specified, the fair saleable value of such share of
         Common Stock (determined without giving effect to the discount for (i)
         a minority interest or (ii) any lack of liquidity of the Common Stock
         or to the fact that the Issuer may have no class of equity registered
         under the Exchange Act) as of the last day of the most recent fiscal
         month to end within 60 days prior to such date specified, based on the
         value of the Issuer, as determined by an investment banking firm
         (selected pursuant to the terms of Section 13 of this Agreement) in
         accordance with such firm's customary practices, divided by the number
         of Outstanding shares of Common Stock, after giving pro forma effect to
         the exercise or conversion of all exercisable or Convertible Securities
         (including the Warrants) for Common Stock and the payment of the
         exercise or conversion price therefor.

         "Book Value" shall mean, in respect of any share of Common Stock on any
         date herein specified, the consolidated book value of the Issuer as of
         the last day of any month immediately preceding such date, divided by
         the number of Outstanding shares of Common Stock, after giving pro
         forma effect to the exercise or conversion of all exercisable or
         Convertible Securities (including the Warrants) for Common Stock and
         the payment of the exercise or conversion price therefor, as determined
         in accordance with GAAP by any firm of independent certified public
         accountants of recognized national standing selected by the Issuer and
         reasonably acceptable to the Required Holders.

         "Business Day" shall mean any day that is not a Saturday or Sunday or a
         day on which banks are required or permitted to be closed in the State
         of New York or California.

         "Cashless Conversion" shall have the meaning set forth in Section
         2.2(b)(ii) hereof.

         "Cashless Conversion Notice" shall have the meaning set forth in
         Section 2.2(b)(ii) hereof.


                                       2
<PAGE>   6

         "Closing Date" shall mean the date hereof.

         "Commission" shall mean the Securities and Exchange Commission, or any
         other federal agency then administering the Securities Act and other
         federal securities laws.

         "Common Stock" shall mean the common stock, $0.01 par value per share,
         of the Issuer, as constituted on the Closing Date, and any capital
         stock into which such Common Stock may thereafter be changed, and shall
         also include (i) capital stock of the Issuer of any other class
         (regardless of how denominated) issued to the holders of shares of
         Common Stock upon any reclassification thereof which is also not
         preferred as to dividends or assets over any other class of stock of
         the Issuer and which is not subject to redemption and (ii) shares of
         common stock of any successor or acquiring corporation (as defined in
         Section 4.8 of this Agreement) received by or distributed to the
         holders of Common Stock of the Issuer in the circumstances contemplated
         by Section 4.8 of this Agreement.

         "Convertible Securities" shall mean evidences of indebtedness, shares
         of stock or other securities which are convertible into or
         exchangeable, with or without payment of additional consideration in
         cash or property, for Additional Shares of Common Stock, either
         immediately or upon the occurrence of a specified date or a specified
         event.

         "Current Market Price" shall mean, in respect of any share of Common
         Stock on any date herein specified, (a) if there shall then be a public
         market for the Common Stock, the average Price per share for the 20
         trading days preceding such date; or (b) at any time that there is no
         public market for the Common Stock, the fair market value per share of
         Common Stock on such date as determined reasonably and in good faith by
         the board of directors of the Issuer (determined without giving effect
         to any discount for a minority interest, any restrictions on
         transferability or any lack of liquidity of the Common Stock or to the
         fact that the Issuer has no class of equity registered under the
         Exchange Act), such fair market value to be determined by reference to
         the cash price that would be paid between a fully informed buyer and
         seller under no compulsion to buy or sell, provided that (i) if Current
         Market Price is being determined in connection with an issuance of
         shares of Common Stock, solely to one or more Affiliates of the Issuer,
         then if so requested by the Required Holders, Current Market Price
         shall be the Appraised Value; and (ii) Current Market Price shall never
         be less than Book Value.

         "Current Warrant Price" shall mean, in respect of a share of Common
         Stock at any date herein specified, two and 25/100 dollars ($2.25) per
         share of Common Stock, subject to adjustment from time to time as
         provided in this Agreement.

         "Demanding Security Holder" shall have the meaning set forth in Section
         9.3.

         "Exchange Act" shall mean the Securities Exchange Act of 1934, as
         amended, or any similar federal statute, and the rules and regulations
         of the Commission thereunder, all as the same shall be in effect from
         time to time.

         "Exercise Period" shall mean the period during which the Warrants are
         exercisable pursuant to Section 2.2.



                                       3
<PAGE>   7

         "Expiration Date" shall mean July 19, 2004.

         "GAAP" shall mean generally accepted accounting principles in the
         United States of America, as from time to time in effect.

         "Material Adverse Effect" shall mean, as to the Issuer, any material
         adverse effect on the business, assets, operations, prospects or
         financial or other condition of the Issuer and its Subsidiaries, taken
         as a whole.

         "NASD" shall mean the National Association of Securities Dealers, Inc.,
         or any successor thereto.

         "Other Property" shall have the meaning set forth in Section 4.8.

         "Outstanding" shall mean, when used with reference to Common Stock, at
         any date as of which the number of shares thereof is to be determined,
         all issued shares of Common Stock, except shares then owned or held by
         or for the account of the Issuer or any Subsidiary, and shall include
         all shares issuable in respect of outstanding scrip or any certificates
         representing fractional interests in shares of Common Stock.

         "Permitted Issuances" shall mean (i) the issuance of shares of Common
         Stock upon exercise of the Warrants, (ii) the issuance of shares
         relating to any benefit plan, stock option plan or any other
         compensation plan offered solely to the Issuer's officers, directors
         and/or employees, (iii) the issuance of shares of Common Stock as
         consideration for the purchase of any property, stock, business or
         securities from any Person who is not an Affiliate of the Issuer or any
         Subsidiary immediately prior to such transaction whether such shares
         are issued directly by the Issuer or a Subsidiary of the Issuer in
         connection with any merger, consolidation or other business
         combination, (iv) if there shall then be a public market for the Common
         Stock, the issuance of shares of Common Stock upon receipt by the
         Issuer of the Current Market Price therefor as described in clause (a)
         of the definition of "Current Market Price" and (v) if there shall then
         be no public market for the Common Stock, the issuance of shares of
         Common Stock, warrants or Convertible Securities on terms that are at
         least as favorable to the Issuer as terms that could be obtained in an
         arm's length transaction with third Persons not Affiliates of the
         Issuer or any Subsidiary and for consideration equal to the fair value
         of such shares as determined in good faith by a majority of
         disinterested members of the board of directors of the Issuer.

         "Person(s)" shall mean any individual, sole proprietorship,
         partnership, joint venture, trust, limited liability company,
         incorporated organization, association, corporation, institution,
         public benefit corporation, entity or government (whether federal,
         state, county, city, municipal or otherwise, including, without
         limitation, any instrumentality, division, agency, body or department
         thereof).

         "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


                                       4
<PAGE>   8

         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. 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.

         "Registrable Securities" shall mean, at any particular time and as to
         each Warrant Holder, (i) all shares of common stock issuable upon the
         exercise of such Warrant Holder's Warrants and (ii) all of such Warrant
         Holder's issued and outstanding Warrant Stock.

         "Registration Expenses" shall have the meaning set forth in Section 9.5
         of this Agreement.

         "Registration Statement" shall have the meaning set forth in Section
         9.4 of this Agreement.

         "Required Holders" shall mean the Warrant Holders of Warrants
         exercisable for an amount exceeding 50% of the aggregate number of
         shares of Common Stock then purchasable upon exercise of all Warrants,
         whether or not exercisable.

         "Requirement of Law" shall mean, as to any Person, any requirement
         contained in any certificate of incorporation, bylaws, or other
         organizational or governing documents of such Person, and any law,
         treaty, rule or regulation or determination of an arbitrator or a court
         or other governmental authority, in each case applicable to or binding
         such Person or any of the property or to which such Person or any of
         its property is subject.

         "Restricted Common Stock" shall mean shares of Common Stock which are,
         or which upon their issuance on the exercise of a Warrant would be,
         evidenced by a certificate bearing the restrictive legend set forth in
         Section 9.1 of this Agreement.

         "Securities Act" shall mean the Securities Act of 1933, as amended, or
         any similar federal statute, and the rules and regulations of the
         Commission thereunder, all as the same shall be in effect at the time.

         "Subsidiary" shall mean any Person of which an aggregate of more than
         50% of the outstanding sock or other ownership interests having
         ordinary voting power to elect a majority of the board of directors or
         other managers of such Person (irrespective of whether, at the time,
         stock or other ownership interests of any other class or classes of
         such Person shall have or might have voting power by reason of the
         happening of any contingency) is at the time, directly or indirectly,
         owned legally or beneficially by the Issuer and/or one or more
         Subsidiaries of the Issuer.

         "Transfer Notice" shall have the meaning set forth in Section 9.2 of
         this Agreement.


                                       5
<PAGE>   9

         "Warrant Certificate" shall mean a certificate evidencing one or more
         Warrants, substantially in the form of Exhibit A hereto, with such
         changes therein as may be required to reflect any adjustments made
         pursuant to Section 4 of this Agreement.

         "Warrant Holder" shall mean such Person in whose name the Warrants are
         registered on the books of the Issuer maintained for such purpose or
         each Person holding any Warrant Stock. As of the Closing Date, the Bank
         of Montreal is the Warrant Holder hereof.

         "Warrant Price" shall mean, for any exercise of Warrants pursuant to
         Section 2.2 of this Agreement, an amount equal to (i) the number of
         shares of Common Stock being purchased upon such exercise multiplied by
         (ii) the Current Warrant Price for each share of Common Stock as of the
         date of such exercise.

         "Warrant Stock" shall mean the shares of Common Stock purchased by the
         Warrant Holders upon the exercise thereof.

         "Warrants" shall mean the Warrants issued pursuant to this Agreement
         and all Warrants issued upon transfer, division or combination of, or
         in substitution for, such Warrants. All Warrants shall at all times be
         identical as to terms and conditions and date, except as to the number
         of shares of Common Stock for which they may be exercised. A Warrant
         shall entitle the record holder thereof to purchase from the Issuer one
         share of Common Stock (subject to adjustment as provided in Section 4
         of this Agreement).

2.       ISSUANCE AND EXERCISE OF WARRANTS

2.1      Issuance of Warrants. The Issuer hereby agrees to issue to the Warrant
         Holder on the Closing Date, SIX HUNDRED SIXTY-ONE THOUSAND FIVE HUNDRED
         THIRTY-EIGHT (661,538) Warrants. On the Closing Date, the Issuer shall
         deliver to the Warrant Holder Warrant Certificates evidencing the
         Warrants issued to the Warrant Holder.

2.2      Manner of Exercise.

         (a)      The Warrant Holder may, from and after the Closing Date until
                  5:00 p.m., New York City time on the Expiration Date, exercise
                  the Warrants evidenced by a Warrant Certificate, on any
                  Business Day, for all or part of the number of shares of
                  Common Stock purchasable thereunder.

         (b)      In order to exercise the Warrants, in whole or in part, the
                  Warrant Holder shall either:

                  (i)      deliver to the Issuer at its principal office at 6300
                           Bridge Point Parkway, Building 2, Suite 500, Austin,
                           Texas 78730, Attention: President, or at the office
                           or agency designated by the Issuer pursuant to
                           Section 12 of this Agreement (the "Principal
                           Office"), (x) a written notice duly executed by the
                           Warrant Holder or its agent or attorney,
                           substantially in the form of the form of election to
                           purchase appearing at the end of the Warrant
                           Certificate as Exhibit A thereto, of such Warrant
                           Holder's election to exercise the Warrants, which
                           notice shall specify the number of shares of


                                       6
<PAGE>   10

                           Common Stock to be purchased, (y) payment of the
                           Warrant Price in the manner provided below, and (z)
                           the Warrant Certificate or Warrant Certificates
                           evidencing the Warrants. Upon receipt thereof, the
                           Issuer shall, as promptly as practicable, and in any
                           event within three (3) Business Days thereafter,
                           execute or cause to be executed and deliver or cause
                           to be delivered to such Warrant Holder a certificate
                           or certificates representing the aggregate number of
                           full shares of Common Stock issuable upon such
                           exercise, together with cash in lieu of any fraction
                           of a share, as hereinafter provided. The stock
                           certificate or certificates so delivered shall be, to
                           the extent possible, in such denomination or
                           denominations as such Warrant Holder shall request in
                           the notice and shall be registered in the name of the
                           Warrant Holder or, subject to Section 9 of this
                           Agreement, such other name as shall be designated in
                           the notice. The Warrants shall be deemed to have been
                           exercised and such certificate or certificates shall
                           be deemed to have been issued, and such Warrant
                           Holder or any other Person so designated to be named
                           therein shall be deemed to have become a holder of
                           record of such shares for all purposes, as of the
                           date the notice, together with payment of the Warrant
                           Price and the Warrant Certificate or Warrant
                           Certificates, are received by the Issuer as described
                           above and all taxes required to be paid by such
                           Warrant Holder, if any, pursuant to Section 2.3 of
                           this Agreement prior to the issuance of such shares
                           have been paid. If the Warrants evidenced by a
                           Warrant Certificate shall have been exercised, the
                           Issuer shall, at the time of delivery of the
                           certificate or certificates representing the Warrant
                           Stock, deliver to the Warrant Holder a new Warrant
                           Certificate evidencing the rights of the Warrant
                           Holder to purchase the unpurchased shares of Common
                           Stock represented by the old Warrant Certificate,
                           which new Warrant Certificate shall in all other
                           respects be identical to the old Warrant Certificate.
                           Payment of the Warrant Price shall be made in cash in
                           an amount equal to the Warrant Price; or

                  (ii)     deliver to the Issuer on any Business Day at the
                           Principal Office the Cashless Conversion Notice in
                           substantially the form attached hereto as Exhibit B
                           duly executed by the Warrant Holder and setting forth
                           such Warrant Holder's election to receive the number
                           of shares of Common Stock specified in the Cashless
                           Conversion Notice ("Cashless Conversion"). Such
                           presentation and surrender shall be deemed a waiver
                           of the Warrant Holder's obligation to pay all or any
                           portion (as the case may be) of the Warrant Price in
                           connection with such Cashless Conversion. In the
                           event of a Cashless Conversion, the Issuer shall
                           deliver to the Warrant Holder (without payment by the
                           Warrant Holder of any Warrant Price) that number of
                           shares of Common Stock equal to: the number of shares
                           of Common Stock for which the Warrant Holder hereof
                           desires to exercise the Warrants multiplied by a
                           fraction, (x) the numerator of which shall be the
                           Current Market Price on the date of such exercise
                           less the Current Warrant Price on the date of such
                           exercise and (y) the denominator of which shall be
                           the Current Market Price on the date of


                                       7
<PAGE>   11

                           such exercise. The number of shares of Common Stock
                           "into which each Warrant is exercisable" shall be one
                           share, subject to adjustment as provided in Section 4
                           hereof. The Warrant Holder may exercise its Cashless
                           Conversion rights, at any time or from time to time,
                           prior to the Expiration Date. The Cashless Conversion
                           Notice shall also specify a place and date not less
                           than one nor more than 20 Business Days from the date
                           of the Cashless Conversion Notice for the closing of
                           such purchase.

2.3      Payment of Taxes. The Issuer shall pay all expenses in connection with,
         and all transfer taxes and other governmental charges that may be
         imposed with respect to, the issuance or delivery of Warrant Stock. The
         Issuer shall not be required, however, to pay any tax or other charge
         imposed in connection with any transfer involved in the issue of any
         certificate for shares of Warrant Stock issuable upon exercise of
         Warrants in any name other than that of Warrant Holder, and in such
         case the Issuer shall not be required to issue or deliver any stock
         certificate until such tax or other charge has been paid or it has been
         established to the satisfaction of the Issuer that no such tax or other
         charge is due.

2.4      Fractional Shares. The Issuer shall not be required to issue a
         fractional share of Common Stock upon the exercise of Warrants as
         provided in Section 2(b)(i) and (ii). As to any fraction of a share
         which the Warrant Holder would otherwise be entitled to purchase upon
         such exercise, the Issuer shall pay a cash adjustment in respect of
         such fraction in an amount equal to the same fraction of the Current
         Market Price per share of Common Stock on the date of exercise. If the
         determination of Current Market Price for purposes of this Section 2.4
         would otherwise require an appraisal to be made by an investment
         banking firm, then Current Market Price for purposes of this Section
         2.4 only shall mean Book Value per share of Common Stock on the date of
         exercise, unless a determination of Appraised Value shall have been
         made within six months prior to such date in which case such Appraised
         Value shall be utilized for the purposes of determining Current Market
         Price.

2.5      Continued Validity. A Warrant Holder of shares of Warrant Stock (other
         than a holder who acquires such shares after the same have been
         publicly sold pursuant to a Registration Statement under the Securities
         Act) shall continue to be entitled with respect to such shares to all
         rights to which it would have been entitled as Warrant Holder under
         Sections 9, 10 and 15 of this Agreement. The Issuer will, at the time
         of each exercise of Warrants or upon the request of the Warrant Holder
         of the shares of Warrant Stock issued upon the exercise thereof,
         acknowledge in writing, in form reasonably satisfactory to such Warrant
         Holder, its continuing obligation to afford to such Warrant Holder all
         such rights; provided, however, that if such Warrant Holder shall fail
         to make any such request, such failure shall not affect the continuing
         obligation of the Issuer to afford to such Warrant Holder all such
         rights.

3.       TRANSFERS, DIVISION AND COMBINATION

3.1      Transfer. Subject to compliance with Section 9 of this Agreement,
         transfer of Warrants, in whole or in part, shall be registered on the
         books of the Issuer to be maintained for such purposes, upon surrender
         of the Warrant Certificate representing such Warrants at


                                       8
<PAGE>   12

         the principal office of the Issuer referred to in Section 2.2 of this
         Agreement or the office or agency designated by the Issuer pursuant to
         Section 12 of this Agreement, together with a written assignment
         substantially in the form of Exhibit B to the Warrant Certificate and a
         written agreement, in form reasonably satisfactory to the Issuer,
         setting forth the new Warrant Holder's agreement to be bound by all of
         the terms of this Agreement each duly executed by the Warrant Holder or
         its agent or attorney, and funds sufficient to pay any transfer taxes
         payable by such Warrant Holder upon the making of such transfer. Upon
         such surrender and, if required, such payment, the Issuer shall,
         subject to Section 9 of this Agreement, 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. A Warrant, if properly assigned in compliance
         with Section 9 of this Agreement, may be exercised by a new Warrant
         Holder for the purchase of shares of Warrant Stock without having a new
         Warrant Certificate or new Warrant Certificates issued.

3.2      Division and Combination. Subject to the provisions of Section 9 of
         this Agreement, any Warrant Certificate may be divided or combined with
         other Warrant Certificates upon presentation thereof at the aforesaid
         office or agency of the Issuer, together with a written notice
         specifying the names and denominations in which new Warrant
         Certificates are to be issued, signed by a Warrant Holder or its agent
         or attorney. Subject to compliance with Section 3.1 of this Agreement
         as to any transfer which may be involved in such division or
         combination, the Issuer shall execute and deliver a new Warrant
         Certificate or Warrant Certificates in exchange for the Warrant
         Certificate or Warrant Certificates to be divided or combined in
         accordance with such notice.

3.3      Expenses. The Issuer shall prepare, issue and deliver at its own
         expense (other than transfer taxes) the new Warrant Certificate or
         Warrant Certificates provided for under this Section 3.

3.4      Maintenance of Books. The Issuer agrees to maintain, at its aforesaid
         office or agency, books for the registration of, and the registration
         of transfer of, the Warrants.

4.       ADJUSTMENTS

         The number of shares of Warrant Stock for which Warrants are
         exercisable, and the price at which such shares may be purchased upon
         exercise of Warrants, shall be subject to adjustment from time to time
         as set forth in this Section 4. The Issuer shall give each Warrant
         Holder notice of any event described below which requires an adjustment
         pursuant to this Section 4 within three (3) Business Days after such
         event.

4.1      Stock Dividends, Subdivisions and Combinations. If at any time the
         Issuer shall:

         (a)      take a record of the holders of its Common Stock for the
                  purpose of entitling them to receive a dividend payable in, or
                  other distribution of, Additional Shares of Common Stock,


                                       9
<PAGE>   13

         (b)      subdivide its outstanding shares of Common Stock into a larger
                  number of shares of Common Stock, or

         (c)      combine its outstanding shares of Common Stock into a smaller
                  number of shares of Common Stock,

                  then (i) the number of shares of Common Stock for which a
                  Warrant is exercisable immediately after the occurrence of any
                  such event shall be adjusted to equal the number of shares of
                  Common Stock which a record holder of the same number of
                  shares of Common Stock for which a Warrant is exercisable
                  immediately prior to the occurrence of such event would own or
                  be entitled to receive after the happening of such event, and
                  (ii) the Current Warrant Price shall be adjusted to equal the
                  Current Warrant Price multiplied by a fraction, the numerator
                  of which shall be the number of shares of Common Stock for
                  which a Warrant is exercisable immediately prior to the
                  adjustment and the denominator of which shall be the number of
                  shares for which a Warrant is exercisable immediately after
                  such adjustment.

4.2      Certain Other Distributions. If at any time the Issuer shall take a
         record of the holders of its Common Stock for the purpose of entitling
         them to receive any dividend or other distribution of:

         (a)      cash;

         (b)      any evidences of its indebtedness (other than Convertible
                  Securities), any shares of its stock (other than Additional
                  Shares of Common Stock or Convertible Securities) or any other
                  securities or property of any nature whatsoever (other than
                  cash); or

         (c)      any warrants or other rights to subscribe for or purchase any
                  evidences of its indebtedness (other than Convertible
                  Securities), any shares of its stock (other than Additional
                  Shares of Common Stock or Convertible Securities) or any other
                  securities or property of any nature whatsoever;

                  then (i) the number of shares of Common Stock for which a
                  Warrant is exercisable shall be adjusted to equal the product
                  obtained by multiplying the number of shares of Common Stock
                  for which a Warrant is exercisable immediately prior to such
                  adjustment by a fraction (A) the numerator of which shall be
                  the Current Market Price per share of Common Stock at the date
                  of taking such record and (B) the denominator of which shall
                  be such Current Market Price per share of Common Stock, minus
                  the amount allocable to one share of Common Stock of any such
                  cash so distributable and of the fair value (as determined
                  reasonably and in good faith by the board of directors of the
                  Issuer) of any and all such evidences of indebtedness, shares
                  of stock, other securities or property or warrants or other
                  subscription or purchase rights so distributable, and (ii) the
                  Current Warrant Price shall be adjusted to equal (A) the
                  Current Warrant Price multiplied by the number of shares of
                  Common Stock for which a Warrant is


                                       10
<PAGE>   14

                  exercisable immediately prior to the adjustment divided by (B)
                  the number of shares for which a Warrant is exercisable
                  immediately after such adjustment. A reclassification of the
                  Common Stock (other than a change in par value, or from par
                  value to no par value or from no par value to par value) into
                  shares of Common Stock and shares of any other class of stock
                  shall be deemed a distribution by the Issuer to the holders of
                  its Common Stock of such shares of such other class of stock
                  within the meaning of this Section 4.2 and, if the Outstanding
                  shares of Common Stock shall be changed into a larger or
                  smaller number of shares of Common Stock as a part of such
                  reclassification, such change shall be deemed a subdivision or
                  combination, as the case may be, of the Outstanding shares of
                  Common Stock within the meaning of Section 4.1 of this
                  Agreement.

4.3      Issuance of Additional Shares of Stock.

         (a)      If at any time the Issuer shall (except as hereinafter
                  provided) issue or sell any Additional Shares of Common Stock,
                  other than Permitted Issuances, for consideration in an amount
                  per Additional Share of Common Stock less than the Current
                  Market Price, then the Current Warrant Price shall be adjusted
                  by multiplying the Current Warrant Price by a fraction, the
                  numerator of which shall be (A) an amount equal to the sum of
                  (X) the number of shares of Common Stock Outstanding
                  immediately prior to such issuance or sale multiplied by the
                  Current Market Price immediately prior to the first to occur
                  of (i) board action by the Issuer authorizing such action or
                  (ii) the public announcement of an intent to take such action,
                  plus (Y) the consideration, if any, received by the Issuer
                  upon such issuance or sale, and the denominator of which shall
                  be (B) the total number of shares of Common Stock Outstanding
                  immediately after such issuance or sale multiplied by the
                  Current Market Price as determined in clause (A) above.

         (b)      The provisions of Section 4.3(a) of this Agreement shall not
                  apply to any issuance of Additional Shares of Common Stock for
                  which an adjustment is provided under Sections 4.1 or 4.2 of
                  this Agreement. No adjustment of the number of shares of
                  Common Stock for which a Warrant shall be exercisable shall be
                  made under Section 4.3(a) of this Agreement upon the issuance
                  of any Additional Shares of Common Stock which are issued
                  pursuant to the exercise of any warrants or other subscription
                  or purchase rights or pursuant to the exercise of any
                  conversion or exchange rights in any Convertible Securities if
                  any such adjustment shall previously have been made upon the
                  issuance of such warrants or other rights or upon the issuance
                  of such Convertible Securities (or upon the issuance of any
                  warrant or other rights therefor) pursuant to Section 4.4 or
                  Section 4.5 of this Agreement.

4.4      Issuance of Warrants or Other Rights. If at any time the Issuer shall
         take a record of the holders of its Common Stock for the purpose of
         entitling them to receive a distribution of, or shall in any manner
         (whether directly or by assumption in a merger in which the Issuer is
         the surviving corporation) issue or sell, any warrants or other rights
         to subscribe for or purchase any Additional Shares of Common Stock or
         any Convertible Securities, whether


                                       11
<PAGE>   15

         or not the rights to exchange or convert thereunder are immediately
         exercisable, and if the price per share for which Common Stock is
         issuable upon the exercise of such warrants or other rights or upon
         conversion or exchange of such Convertible Securities shall be less
         than the Current Market Price in effect immediately prior to the time
         of such distribution, issue or sale, then the Current Warrant Price
         shall be adjusted as provided in Section 4.3(a) of this Agreement on
         the basis that (A) the maximum number of Additional Shares of Common
         Stock issuable pursuant to all such warrants or other rights or
         necessary to effect the conversion or exchange of all such Convertible
         Securities shall be deemed to be Outstanding immediately following such
         issuance, (B) the price per share for such Additional Shares of Common
         Stock shall be deemed to be the lowest possible price per share in any
         range of prices per share at which such Additional Shares of Common
         Stock are available to such holders, and (C) the Issuer shall be deemed
         to have received all of the consideration payable therefor, if any, as
         of the date of the actual issuance of such warrants or other rights. No
         further adjustments of the Current Warrant Price shall be made upon the
         actual issuance of such Common Stock or of such other rights or upon
         exercise of such warrants or other rights or upon the actual issuance
         of such Common Stock upon such conversion or exchange of such
         Convertible Securities.

4.5      Issuance of Convertible Securities. If at any time the Issuer shall
         take a record of the holders of its Common Stock for the purpose of
         entitling them to receive a distribution of, or shall in any manner
         (whether directly or by assumption in a merger in which the Issuer is
         the surviving corporation) issue or sell, any Convertible Securities,
         whether or not the rights to exchange or convert thereunder are
         immediately exercisable, and if the price per share for which Common
         Stock is issuable upon such conversion or exchange shall be less than
         the Current Market Price in effect immediately prior to the time of
         such issue or sale of Convertible Securities, then the Current Warrant
         Price shall be adjusted as provided in Section 4.3(a) of this Agreement
         on the basis that (A) the maximum number of Additional Shares of Common
         Stock necessary to effect the conversion or exchange of all such
         Convertible Securities shall be deemed to be Outstanding immediately
         following such issuance, (B) the price per share of such Additional
         Shares of Common Stock shall be deemed to be the lowest possible price
         in any range of prices at which such Additional Shares of Common Stock
         are available to such holders, and (C) the Issuer shall be deemed to
         have received all of the consideration payable therefor, if any, as of
         the date of actual issuance of such Convertible Securities. No
         adjustment of the Current Warrant Price shall be made under this
         Section 4.5 upon the issuance of any Convertible Securities which are
         issued pursuant to the exercise of any warrants or other subscription
         or purchase rights therefor if any such adjustments shall previously
         have been made upon the issuance of such warrants or other rights
         pursuant to Section 4.4 of this Agreement. No further adjustments of
         the Current Warrant Price shall be made upon the actual issue of such
         Common Stock upon conversion or exchange of such Convertible Securities
         and, if any issue or sale of such Convertible Securities is made upon
         exercise of any warrant or other right to purchase any such Convertible
         Securities for which adjustments of the Current Warrant Price have been
         or are to be made pursuant to other provisions of this Section 4, no
         further adjustments of the Current Warrant Price shall be made by
         reason of such issue or sale.


                                       12
<PAGE>   16

4.6      Superseding Adjustment. If, at any time after any adjustment of the
         Current Warrant Price shall have been made pursuant to Section 4.4 or
         Section 4.5 of this Agreement as the result of any issuance of
         warrants, options, rights or Convertible Securities, and such warrants,
         options or rights, or the right of conversion or exchange in such other
         Convertible Securities, shall expire, and all or a portion of such
         warrants, options or rights, or the right of conversion or exchange
         with respect to all or a portion of such other Convertible Securities,
         as the case may be, shall not have been exercised, then such previous
         adjustment shall be rescinded and annulled and, if applicable, the
         Current Warrant Price shall be recalculated as if all such expired and
         unexercised warrants, options, rights or Convertible Securities had
         never been issued.

4.7      Other Provisions Applicable to Adjustments Under This Section. The
         following provisions shall be applicable to the making of adjustments
         of the number of shares of Common Stock for which a Warrant is
         exercisable provided for in this Section 4:

         (a)      Computation of Consideration. To the extent that any
                  Additional Shares of Common Stock shall be issued for cash
                  consideration, the consideration received by the Issuer
                  therefor shall be the amount of the cash received by the
                  Issuer therefor, or, if such Additional Shares of Common Stock
                  are sold to underwriters or dealers for public offering
                  without a subscription offering, the initial public offering
                  price (in any such case subtracting any amounts paid or
                  receivable for accrued interest or accrued dividends, but not
                  subtracting any compensation, discounts or expenses paid or
                  incurred by the Issuer for and in the underwriting of, or
                  otherwise in connection with, the issuance thereof). To the
                  extent that such issuance shall be for a consideration other
                  than cash, then, except as herein otherwise expressly
                  provided, the amount of such consideration shall be deemed to
                  be the fair value of such consideration at the time of such
                  issuance as determined reasonably and in good faith by a
                  majority of the disinterested members of the board of
                  directors of the Issuer.

         (b)      When Adjustments to Be Made. The adjustments required by this
                  Section 4 shall be made whenever and as often as any specified
                  event requiring an adjustment shall occur, except that any
                  adjustment to the number of shares for which the Warrants are
                  exercisable that would otherwise be required may be postponed
                  (except in the case of a subdivision or combination of shares
                  of the Common Stock, as provided for in Section 4.1 of this
                  Agreement) up to, but not beyond, the date and time of
                  exercise of any Warrants if such adjustment either by itself
                  or with other adjustments not previously made adds or
                  subtracts less than 1% to the number of shares of Common Stock
                  for which the Warrants initially issued pursuant to this
                  Agreement are exercisable immediately prior to the making of
                  such adjustment. Any adjustment representing a change of less
                  than such minimum amount (except as aforesaid) which is
                  postponed shall be carried forward and made as soon as such
                  adjustment, together with other adjustments required by this
                  Section 4 and not previously made, would result in a minimum
                  adjustment or on the date of exercise. For the purpose of any
                  adjustment, any specified event shall be deemed to have
                  occurred at the close of business on the date of its
                  occurrence.



                                       13
<PAGE>   17

         (c)      Fractional Interests. In computing adjustments under this
                  Section 4, fractional interests in Common Stock resulting from
                  an issuance of additional Warrants to any Warrant Holder
                  pursuant to this Section 4 shall be taken into account to the
                  nearest 1/10th of a share, subject to Section 2.4 of this
                  Agreement.

         (d)      When Adjustment Not Required. If the Issuer shall take a
                  record of the holders of its Common Stock for the purpose of
                  entitling them to receive a dividend or distribution or
                  subscription or purchase rights and shall, thereafter and
                  before the distribution to stockholders thereof, legally
                  abandon its plan to pay or deliver such dividend,
                  distribution, subscription or purchase rights, no adjustment
                  shall be required by reason of the taking of such record and
                  any such adjustment previously made in respect thereof shall
                  be rescinded and annulled.

         (e)      Escrow of Warrant Stock. If after any property becomes
                  distributable pursuant to this Section 4 by reason of the
                  taking of any record of the holders of Common Stock, but prior
                  to the occurrence of the event for which such record is taken,
                  any Warrant Holder exercises Warrants, any Additional Shares
                  of Common Stock issuable upon exercise of such Warrant by
                  reason of such adjustment shall be deemed the last shares of
                  Common Stock for which such Warrant is exercised
                  (notwithstanding any other provision to the contrary herein),
                  and such shares or other property shall be held in escrow for
                  a Warrant Holder by the Issuer to be issued to such Warrant
                  Holder upon and to the extent that the event actually takes
                  place, upon payment of the balance, if any, of the Warrant
                  Price for such Warrant at such date (after taking into account
                  any overpayment of the Warrant Price made at any time of the
                  initial Warrant exercise). Notwithstanding any other provision
                  to the contrary herein, if the event for which such record was
                  taken fails to occur or is rescinded, then such escrowed
                  shares shall be canceled by the Issuer and escrowed property
                  returned.

4.8      Reorganization, Reclassification, Merger, Consolidation or Disposition
         of Assets. In the event the Issuer shall reorganize its capital,
         reclassify its capital stock, consolidate or merge with and into
         another corporation or entity (where the Issuer is not the surviving
         corporation or where there is a change in or distribution with respect
         to the Common Stock of the Issuer), or sell, transfer or otherwise
         dispose of all or substantially all its property, assets or business to
         another corporation or entity and, pursuant to the terms of such
         reorganization, reclassification, merger, consolidation or disposition
         of assets, shares of common stock of the successor or acquiring
         corporation or entity, or any cash, shares of stock or other securities
         or property of any nature whatsoever (including warrants or other
         subscription or purchase rights) in addition to or in lieu of common
         stock of the successor or acquiring corporation or entity ("Other
         Property"), are to be received by or distributed to the holders of
         Common Stock of the Issuer, then the Issuer shall, as a condition
         precedent to such transaction, cause effective provisions to be made so
         that each Warrant Holder shall have the right thereafter to receive,
         upon exercise of a warrant, solely the number of shares of "common
         stock of the successor or acquiring corporation" or of the Issuer, if
         it is the surviving corporation, and Other Property receivable upon or
         as a result of such reorganization, reclassification, merger,
         consolidation or disposition of assets, by a holder of the number of
         shares of Common


                                       14
<PAGE>   18

         Stock for which a Warrant is exercisable immediately prior to such
         event. In case of any such reorganization, reclassification, merger,
         consolidation or disposition of assets, such provisions shall include
         the express assumption by the successor or acquiring corporation or
         entity (if other than the Issuer) of the due and punctual observance
         and performance of each and every covenant and condition of this
         Agreement to be performed and observed by the Issuer and all the
         obligations and liabilities hereunder, subject to such modifications as
         may be deemed appropriate (as determined by resolution of the board of
         directors of the Issuer) to provide for adjustments of shares of the
         Common Stock for which a Warrant is exercisable which shall be as
         nearly equivalent as practicable to the adjustments provided for in
         this Section 4. For purposes of this Section 4.8, "common stock of the
         successor or acquiring corporation" shall include stock of such
         corporation of any class which is not preferred as to dividends or
         assets over any other class of stock or other securities of such
         corporation or entity and which is not subject to redemption and shall
         also include any evidences of indebtedness, shares of stock or other
         securities which are convertible into or exchangeable for any such
         stock or other securities, either immediately or upon the arrival of a
         specified date or the happening of a specified event, and any warrants
         or other rights to subscribe for or purchase any such stock or
         securities. The foregoing provisions of this Section 4.8 shall
         similarly apply to successive reorganizations, reclassifications,
         mergers, consolidations or disposition of assets.

5.       NOTICES TO WARRANT HOLDERS

5.1      Notice of Adjustments. Whenever the number of shares of Common Stock
         for which a Warrant is exercisable, or whenever the price at which a
         share of such Common Stock may be purchased upon exercise of the
         Warrants, shall be adjusted pursuant to Section 4, the Issuer shall
         forthwith prepare a certificate to be executed by the chief financial
         officer of the Issuer setting forth, in reasonable detail, the event
         requiring the adjustment and the method by which such adjustment was
         calculated (including a description of the basis on which the board of
         directors of the Issuer determined the fair value of any evidences of
         indebtedness, shares of stock, other securities or property or warrants
         or other subscription or purchase rights referred to in Section 4 of
         this Agreement), specifying the number of shares of Common Stock for
         which a Warrant is exercisable and (if such adjustment was made
         pursuant to Section 4.8 of this Agreement) describing the number and
         kind of any other shares of stock or Other Property for which a Warrant
         is exercisable, and any change in the purchase price or prices thereof,
         after giving effect to such adjustment or change. The Issuer shall
         promptly cause a signed copy of such certificate to be delivered to
         each Warrant Holder in accordance with Section 15.2 of this Agreement.
         The Issuer shall keep at its office or agency designated pursuant to
         Section 12 of this Agreement copies of all such certificates and cause
         the same to be available for inspection at said office during normal
         business hours by any Warrant Holder or any prospective purchaser of a
         Warrant designated by a Warrant Holder thereof.

5.2      Notice of Certain Corporate Action. Each Warrant Holder shall be
         entitled to the same rights to receive notice of corporate action as
         any holder of Common Stock.


                                       15
<PAGE>   19

6.       REPRESENTATIONS AND WARRANTIES

         The Issuer makes the following representations and warranties, each and
         all of which shall be true and correct as of the date of execution and
         delivery of this Agreement and shall survive the execution and delivery
         of this Agreement:

         (a)      Due Organization; Etc. The Issuer is a corporation duly
                  organized validly existing and in good standing under the laws
                  of the State of Delaware, and has the power and authority to
                  execute and deliver this Agreement and the Warrant
                  Certificates, to issue the Warrants and to perform its
                  obligations under this Agreement and the Warrant Certificates.

         (b)      Due Authorization; No Violation. The execution, delivery and
                  performance by the Issuer of this Agreement and the Warrant
                  Certificates, the issuance of the Warrants and the issuance of
                  the Warrant Stock upon exercise of the Warrants have been duly
                  authorized by all necessary corporate action and do not and
                  will not violate, or result in a breach of, or constitute a
                  default under or require any consent under, or result in the
                  creation of any lien or security interest upon the assets of
                  the Issuer pursuant to, any Requirement of Law or any
                  contractual obligation binding upon the Issuer.

         (c)      Due Execution; Etc. This Agreement has been duly executed and
                  delivered by the Issuer and constitutes a legal, valid and
                  enforceable obligation of the Issuer. When the Warrants and
                  the Warrant Certificates have been issued as contemplated
                  hereby, (i) the Warrants and the Warrant Certificates will
                  constitute legal, valid, binding and enforceable obligations
                  of the Issuer and (ii) the Warrant Stock, when issued upon
                  exercise of the Warrants in accordance with the terms hereof,
                  will be duly authorized, validly issued, fully paid and
                  non-assessable shares of Common Stock with no personal
                  liability attaching to the ownership thereof.

         (d)      Capitalization. The total number of shares of all classes of
                  stock that the Issuer shall on the Closing Date have authority
                  to issue is 40,000,000 shares, consisting of (i) 30,000,000
                  shares of Common Stock, par value $0.01 per share, of which,
                  after giving effect to the transactions contemplated herein
                  and all other issuances of capital stock of the Issuer on or
                  prior to the Closing Date, 14,428,621 shares of Common Stock
                  will be issued and outstanding and 661,538 shares of Common
                  Stock will be reserved for future issuance pursuant to this
                  Agreement and (ii) 10,000,000 shares of Preferred Stock, par
                  value $0.01 per share, none of which are presently
                  outstanding. Schedule A sets forth a complete list of the
                  outstanding capital stock of the Issuer, including any
                  options, warrants or rights to purchase the capital stock of
                  the Issuer. The delivery hereunder by the Issuer to the
                  Warrant Holder of the Warrants issued on the Closing Date will
                  transfer and convey to the Warrant Holder good and marketable
                  title to such Warrants and, upon exercise of such Warrants in
                  accordance with this Agreement, good and marketable title to
                  the Common Stock purchased upon such exercise, free and clear
                  of all preemptive rights, liens, charges and encumbrances,
                  except for


                                       16
<PAGE>   20

                  restrictions on transfer referred to in this Agreement, or
                  arising under the Federal and state securities laws. Except as
                  otherwise disclosed on Schedule A, the Issuer does not have
                  outstanding any stock or securities convertible into or
                  exchangeable for any shares of its stock, nor, except as so
                  set forth, does it have outstanding any agreements, rights or
                  options entitling any person to subscribe for or to purchase
                  any capital stock or securities convertible into or
                  exchangeable for any of its shares of stock.

         (e)      Full Disclosure. No information contained in this Agreement,
                  the financial statements referred to in the Credit Agreement
                  or any written statement furnished by or on behalf of the
                  Issuer pursuant to the terms of this Agreement to the Warrant
                  Holder contains any untrue statement of a material fact or
                  omits to state a material fact necessary to make the
                  statements contained herein or therein not misleading in light
                  of the circumstances under which made.

         (f)      Warrant Price. The Issuer has taken all corporate action, and
                  obtained all necessary authorizations or exemptions from any
                  public regulatory body or bodies or governmental entity or
                  entities having jurisdiction thereof, as may be necessary in
                  order that the Issuer may validly and legally issue fully paid
                  and non-assessable shares of Common Stock upon to exercise of
                  the warrants at the Warrant Price, as the same may be adjusted
                  pursuant hereto.

         (g)      Other Representations and Warranties. The Issuer hereby
                  affirms and reaffirms for the express benefit of the Warrant
                  Holders that the representations and warranties made by the
                  Issuer in that certain Guaranty Agreement dated as of January
                  26, 1998, as amended, are true and correct, as if made in
                  favor of the Warrant Holder on the date hereof.

7.       CERTAIN COVENANTS

7.1      No Impairment. The Issuer shall not by any action including, without
         limitation, amending its certificate of incorporation or through any
         reorganization, transfer of assets, consolidation, merger, dissolution,
         issue or sale of securities or any other voluntary action, avoid or
         seek to avoid the observance or performance of any of the terms of this
         Agreement, but will at all times in good faith assist in the carrying
         out of all such terms and in the taking of all such actions as may be
         necessary or appropriate to protect the rights of each Warrant Holder
         against impairment. Without limiting the generality of the foregoing,
         the Issuer will use reasonable good faith efforts to obtain all such
         authorizations, exemptions or consents from any public regulatory body
         having jurisdiction thereof as may be necessary to enable it to perform
         its obligations under this Agreement.

         Upon the request of a Warrant Holder, the Issuer will, at any time
         during the period this Agreement is in effect, acknowledge in writing,
         in form satisfactory to such Warrant Holder, the continuing validity of
         this Agreement and the obligations of the Issuer hereunder.


                                       17
<PAGE>   21

7.2      Reservation and Authorization of Common Stock; Registration with, or
         Approval of, any Governmental Authority. From and after the Closing
         Date, the Issuer shall at all times reserve and keep available for
         issue upon the exercise of Warrants such number of its authorized but
         unissued shares of Common Stock as will be sufficient to permit the
         exercise in full of all outstanding Warrants. All shares of Common
         Stock which shall be so issuable, when issued upon exercise of any
         Warrants and payment therefor in accordance with the terms of this
         Agreement, shall be duly and validly issued and fully paid and
         non-assessable, and not subject to preemptive rights.

         Before taking any action which would result in an adjustment in the
         number of shares of Common Stock for which a Warrant is exercisable or
         in the Current Warrant Price, the Issuer shall obtain all such
         authorizations or exemptions thereof, or consents thereto, as may be
         necessary from any public regulatory body or bodies or governmental
         entity or entities having jurisdiction thereof.

         If any shares of Common Stock required to be reserved for issuance upon
         exercise of Warrants require registration or qualification with any
         governmental authority under any federal or state law (otherwise than
         as provided in Section 9 of this Agreement) before such shares may be
         so issued, the Issuer will in good faith and as expeditiously as
         possible and at its expense endeavor to cause such shares to be duly
         registered.

8.       TAKING OF RECORD; STOCK AND WARRANT TRANSFER BOOKS

         In the case of all dividends or other distributions by the Issuer to
         the holders of its Common Stock with respect to which any provision of
         Section 4 of this Agreement refers to the taking of a record of such
         holders, the Issuer will in each such case take such a record as of the
         close of business on a Business Day. The Issuer will not at any time,
         except upon dissolution, liquidation or winding up of the Issuer, close
         its stock transfer books or Warrant transfer books so as to result in
         preventing or delaying the exercise or transfer of any Warrants.

9.       RESTRICTIONS ON TRANSFERABILITY

         The Warrants and the Warrant Stock shall not be transferred before
         satisfaction of the conditions specified in this Section 9, which
         conditions are intended to ensure compliance with the provisions of the
         Securities Act and applicable state securities laws with respect to the
         transfer of any Warrant or any Warrant Stock. Each Warrant Holder, by
         entering into this Agreement and accepting the Warrants, agrees to be
         bound by the provisions of this Section 9.

9.1      Restrictive Legend. Except as otherwise provided in this Section 9,
         each certificate representing Warrants or Warrant Stock, shall be
         stamped or otherwise imprinted with a legend in substantially the
         following form:

         "THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN
         REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR ANY
         APPLICABLE STATE SECURITIES LAWS, AND MAY NOT BE SOLD OR TRANSFERRED IN
         THE ABSENCE OF SUCH REGISTRATION OR AN


                                       18
<PAGE>   22

         EXEMPTION THEREFROM. SUCH SECURITIES ARE SUBJECT TO THE RESTRICTIONS
         AND PRIVILEGES SPECIFIED IN A WARRANT AGREEMENT, DATED AS OF JULY 19,
         1999, BETWEEN BRIGHAM EXPLORATION COMPANY AND THE INITIAL HOLDERS OF
         SECURITIES NAMED THEREIN, A COPY OF WHICH IS ON FILE WITH THE SECRETARY
         OF BRIGHAM EXPLORATION COMPANY AND WILL BE FURNISHED WITHOUT CHARGE TO
         THE HOLDER HEREOF UPON WRITTEN REQUEST, AND THE HOLDER OF THIS
         CERTIFICATE AGREES TO BE BOUND THEREBY."

9.2      Notice of Proposed Transfers; Requests for Registration. Prior to any
         transfer of any Warrants or any shares of Restricted Common Stock, the
         Warrant Holder of such Warrants or Restricted Common Stock shall give
         five days prior written notice to the Issuer of such Warrant Holder's
         intention to effect such transfer (a "Transfer Notice"). Each Warrant
         Holder agrees that it will not sell, transfer or otherwise dispose of
         Warrants or any shares of Restricted Common Stock, in whole or in part,
         except pursuant to an effective registration statement under the
         Securities Act or an exemption from registration thereunder. Each
         certificate, if any, evidencing such shares of Restricted Common Stock
         issued upon such transfer shall bear the restrictive legend set forth
         in Section 9.1, and each Warrant Certificate issued upon such transfer
         shall bear the restrictive legend set forth in Section 9.1 of this
         Agreement, unless in the opinion of the transferee's or Warrant
         Holder's counsel delivered to the Issuer in connection with such
         transfer such legend is not required in order to ensure compliance with
         the Securities Act.

         The Warrant Holders of Warrants and Warrant Stock shall have the right
         to request registration of such Warrant Stock pursuant to Section 9.3
         of this Agreement.

9.3      Incidental Registration. If the Issuer at any time proposes to file on
         its behalf and/or on behalf of any of its security holders (the
         "Demanding Security Holders") a Registration Statement under the
         Securities Act on any form (other than a Registration Statement (i)
         filed pursuant to demand under the Company's Registration Rights
         Agreement with Joint Energy Development Investments II Limited
         Partnership, a Delaware limited partnership, and Enron Capital & Trade
         Resources Corp., a Delaware corporation, dated August 20, 1998, as
         amended, or (ii) on Form S-8 or any similar or successor form or any
         other registration statement relating to an offering of securities
         solely to the Issuer's existing security holders or employees) to
         register the offer and sale of its Common Stock for cash, it will give
         written notice to all Warrant Holders of Warrants or Warrant Stock at
         least twenty (20) days before the anticipated date of initial filing
         with the Commission of such Registration Statement, which notice shall
         set forth the Issuer's intention to effect such a registration, the
         class or series and number of equity securities proposed to be
         registered and the intended method of disposition of the securities
         proposed to be registered by the Issuer. The notice shall offer to
         include in such filing all of the Warrant Holder's Registrable
         Securities.

         Each Warrant Holder desiring to have Registrable Securities registered
         under this Section 9.3 shall advise the Issuer in writing within
         fifteen (15) days after the date of receipt of such offer from the
         Issuer, setting forth the amount of such Registrable


                                       19
<PAGE>   23

         Securities for which registration is requested. The Issuer shall
         thereupon include in such filing the number of shares of Registrable
         Securities for which registration is so requested, subject to the next
         sentence, and shall use its best efforts to effect registration under
         the Securities Act of such securities. If the managing underwriter of a
         proposed public offering shall advise the Issuer in writing that, in
         its opinion, the distribution of the Registrable Securities requested
         to be included in the registration concurrently with the securities
         being registered by the Issuer or any Demanding Security Holder would
         materially and adversely affect the distribution of such securities by
         the Issuer or such Demanding Security Holders, then all selling
         security holders (but not the Issuer or the Demanding Security Holders)
         shall reduce the amount of securities each intended to distribute
         through such offering on a pro rata basis to the greatest aggregate
         amount which, in the opinion of such managing underwriter, would not
         materially and adversely affect the distribution of such securities.

         Nothing in this Section 9.3 shall preclude the Issuer from
         discontinuing the registration of its securities being effected on its
         behalf under this Section 9.3 at any time prior to the effective date
         of the registration relating thereto. Notwithstanding any provision
         herein, the rights of the Warrant Holder under this Section 9.3 are
         subject to the express limitations contained in registration rights
         agreements in effect on the date hereof between the Issuer and other
         parties; provided, however, that the Issuer shall not on or after the
         date of this Agreement enter into any registration rights agreement
         with respect to its securities that conflict with the registration
         rights granted to the Warrant Holder herein.

9.4      Registration Procedures. If the Issuer is required by the provisions of
         this Section 9 to use its best efforts to effect the registration of
         any of its securities under the Securities Act, the Issuer will, as
         expeditiously as possible:

         (a)      prepare and file with the Commission a registration statement
                  with respect to such securities (a "Registration Statement")
                  and use its best efforts to cause such Registration Statement
                  to become and remain effective for the period described in
                  paragraph (b) below;

         (b)      prepare and file with the Commission such amendments and
                  supplements to such Registration Statement and the prospectus
                  used in connection therewith as may be necessary to keep such
                  Registration Statement effective and to comply with the
                  provisions of the Securities Act with respect to the sale or
                  other disposition of all securities covered by such
                  Registration Statement until the earlier of such time as all
                  of such securities have been disposed of in a public offering
                  or the expiration of 90 days;

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


                                       20
<PAGE>   24

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

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

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

                  It shall be a condition precedent to the obligation of the
                  Issuer to take any action pursuant to this Section 9 in
                  respect of the securities which are to be registered at the
                  request of any Warrant Holder of Registrable Securities that
                  such Warrant Holder shall furnish to the Issuer such
                  information regarding the securities held by such Warrant
                  Holder and the intended method of disposition thereof as the
                  Issuer shall reasonably request and as shall be required in
                  connection with the action taken by the Issuer.

9.5      Expenses. All expenses incurred in complying with this Section 9,
         including, without limitation, all registration and filing fees
         (including all expenses incident to filing with the NASD), printing
         expenses, fees and disbursements of counsel for the Issuer, the
         reasonable fees and expenses of one counsel for the selling security
         holders (selected by the Person holding the plurality of the securities
         being registered), expenses of any special audits incident to or
         required by any such registration and expenses of complying with the
         securities or blue sky laws of any jurisdictions pursuant to Section
         9.4(d) of this Agreement (all of such expenses shall be collectively
         referred to herein as "Registration Expenses"), shall be paid by the
         Issuer; provided, however, the Issuer shall not be responsible for any
         discount or commission or cost reimbursement to any underwriter in
         respect of the securities sold by such Warrant Holder of Registrable
         Securities.

9.6      Indemnification and Contribution.

         (a)      In the event of any registration of any of the Registrable
                  Securities under the Securities Act pursuant to this Section
                  9, the Issuer shall indemnify and hold harmless each Warrant
                  Holder of such Registrable Securities, such Warrant


                                       21
<PAGE>   25

                  Holder's directors and officers, each Affiliate of such
                  Warrant Holder, and each other Person (including each
                  underwriter) who participated in the offering of such
                  Registrable Securities and each other Person, if any, who
                  controls such Warrant Holder or such participating Person, if
                  any, who controls such Warrant Holder or such participating
                  Person within the meaning of the Securities Act, against any
                  losses, claims, damages or liabilities, joint or several, to
                  which such Warrant Holder or any such director or officer or
                  participating Person or Affiliate or controlling Person may
                  become subject under the Securities Act or any other statute
                  or at common law, insofar as such losses, claims, damages or
                  liabilities (or actions in respect thereof) arise out of or
                  are based upon (i) any alleged untrue statement of any
                  material fact contained, on the effective date thereof, in any
                  Registration Statement under which such securities were
                  registered under the Securities Act, any preliminary
                  prospectus or final prospectus contained therein, or any
                  amendment or supplement thereto, or (ii) any alleged omission
                  to state therein a material fact required to be stated therein
                  or necessary to make the statements therein, in light of the
                  circumstances under which they were made, not misleading, and
                  shall reimburse such Warrant Holder or such director, officer
                  or participating Person or Affiliate or controlling Person for
                  any legal or any other expenses reasonably incurred by such
                  Warrant Holder or such director, officer or participating
                  Person or Affiliate or controlling Person in connection with
                  investigating or defending any such loss, claim, damage,
                  liability or action; provided, however, that the Issuer shall
                  not be liable in any such case to the extent that any such
                  loss, claim, damage or liability arises out of or is based
                  upon any alleged untrue statement or alleged omission made in
                  such Registration Statement, preliminary prospectus,
                  prospectus or amendment or supplement in reliance upon and in
                  conformity with written information furnished to the Issuer by
                  such Warrant Holder specifically for use therein and, in the
                  case of any non-underwritten offering, to the extent that any
                  such losses, claims, damages, liabilities or expenses arise
                  out of or are based upon the fact that a current copy of the
                  prospectus was not sent or given to the Person asserting any
                  such losses, claims, damages, liabilities or expenses at or
                  prior to the written confirmation of the sale of the
                  securities to such Person if it is determined that it was the
                  responsibility of such Warrant Holder to provide such Person
                  with a current copy of the prospectus and such current copy of
                  the prospectus would have cured the defect giving rise to such
                  losses, claims, damages, liabilities or expenses. Such
                  indemnity shall remain in full force and effect regardless of
                  any investigation made by or on behalf of such Warrant Holder
                  or such director, officer or participating Person or Affiliate
                  or controlling Person, and shall survive the transfer of such
                  securities by such Warrant Holder.

         (b)      Each Warrant Holder of any Registrable Securities, by
                  acceptance thereof, agrees to indemnify and hold harmless the
                  Issuer, its directors and officers and each other Person, if
                  any, who controls the Issuer within the meaning of the
                  Securities Act against any losses, claims, damages or
                  liabilities, joint or several, to which the Issuer or any such
                  director or officer or any such Person may become subject
                  under the Securities Act or any other statute or at common
                  law, insofar as such losses, claims, damages or liabilities
                  (or actions in respect thereof) arise out of or


                                       22
<PAGE>   26

                  are based upon (i) information in writing provided to the
                  Issuer by such Warrant Holder of such Registrable Securities
                  contained, on the effective date thereof, in any Registration
                  Statement under which securities were registered under the
                  Securities Act at the request of such Warrant Holder, any
                  preliminary prospectus or final prospectus contained therein,
                  or any amendment or supplement thereto or (ii) the fact that a
                  current copy of the prospectus was not sent to the Person
                  asserting such losses, claims, damages, liabilities or
                  expenses at or prior to the written confirmation of the sale
                  of the securities with respect to such Person if it is
                  determined that it was the responsibility of such Warrant
                  Holder to provide such Person with a current copy of the
                  prospectus and such current copy would have cured the defect
                  giving rise to such losses, claims, damages, liabilities or
                  expenses; provided, however, that such Warrant Holder's
                  obligation under this Section 9.6(b) to indemnify and hold
                  harmless the Issuer shall in no event exceed the lesser of (x)
                  the damage attributable solely to the inclusion of such
                  written information in such Registration Statement,
                  preliminary prospectus, final prospectus, or amendment or
                  supplement suffered by the Person or Persons whose claims gave
                  rise to such losses, claims, damages or liabilities and (y)
                  the net proceeds received by such Warrant Holder from the sale
                  of Registrable Securities giving rise to such indemnification.

         (c)      If the indemnification provided for in this Section 9 from the
                  indemnifying party is unavailable to an indemnified party
                  hereunder in respect of any losses, claims, damages,
                  liabilities or expenses referred to herein, then the
                  indemnifying party, in lieu of indemnifying such indemnified
                  party, shall contribute to the amount paid or payable by such
                  indemnified party as a result of such losses, claims, damages,
                  liabilities or expenses in such proportion as is appropriate
                  to reflect the relative fault of the indemnifying party and
                  indemnified parties in connection with the actions which
                  resulted in such losses, claims, damages, liabilities or
                  expenses, as well as any other relevant equitable
                  considerations. The relative fault of such indemnifying party
                  and indemnified parties shall be determined by reference to,
                  among other things, whether any action in question, including
                  any untrue or alleged untrue statement of a material fact or
                  omission or alleged omission to state a material fact, has
                  been made by, or related to information supplied by, such
                  indemnifying party or indemnified parties, and the parties'
                  relative intent, knowledge, access to information and
                  opportunity to correct or prevent such action. The amount paid
                  or payable by a party under this Section 9 as a result of the
                  losses, claims, damages, liabilities and expenses referred to
                  above shall be deemed to include any legal or other fees or
                  expenses reasonably incurred by such party in connection with
                  any investigation or proceeding.

                           The parties hereto agree that it would not be just
                  and equitable if contribution pursuant to this Section 9.6(c)
                  were determined by pro rata allocation or by any other method
                  of allocation which does not take account of the equitable
                  considerations referred to in the immediately preceding
                  paragraph. Notwithstanding the provisions of this subsection
                  (c), no Warrant Holder shall be required to contribute any
                  amount in excess of the total amount received by it upon the
                  sale of its securities pursuant to the Registration Statement
                  to which the


                                       23
<PAGE>   27

                  losses, claims, damages, liabilities and expenses referred to
                  above relate. No Person guilty of fraudulent misrepresentation
                  (within the meaning of Section 11(f) of the Securities Act)
                  shall be entitled to contribution from any Person who was not
                  guilty of such fraudulent misrepresentation. The obligations
                  of each of the Warrant Holders under this subsection (c) to
                  contribute are several and not joint.

         (d)      Conduct of Indemnification Proceedings. Any person or entity
                  entitled to indemnification hereunder shall (i) give prompt
                  written notice to the indemnifying party after the receipt by
                  the indemnified party of a written notice of the commencement
                  of any action, suit, proceeding or investigation or threat
                  thereof made in writing for which such indemnified party will
                  claim indemnification or contribution pursuant to this
                  Agreement; provided, however, that the failure of any
                  indemnified party to give notice as provided herein shall not
                  relieve the indemnifying party of its obligations under
                  Section 9.6 hereof, except to the extent that the indemnifying
                  party is actually prejudiced by such failure to give notice,
                  and (ii) unless in such indemnified party's reasonable
                  judgment a conflict of interest may exist between such
                  indemnified and indemnifying parties with respect to such
                  claim, permit such indemnifying party to assume the defense of
                  such claim with counsel reasonably satisfactory to the
                  indemnified party. If the indemnifying party is entitled to,
                  and does, assume the defense of such claim, the indemnified
                  party shall have the right to employ separate counsel and to
                  participate in the defense thereof, but the fees and expenses
                  of such counsel shall be borne by the indemnified party.
                  Whether or not such defense is assumed by the indemnifying
                  party, the indemnifying party shall not be subject to any
                  liability for any settlement made without its consent (but
                  such consent will not be unreasonably withheld). No
                  indemnifying party shall be permitted to consent to the entry
                  of any judgment or to enter into any settlement that does not
                  include as an unconditional term thereof the giving by the
                  claimant or plaintiff to such indemnified party of a release
                  from all liability in respect of such claim or litigation. An
                  indemnifying party who is not entitled to, or elects not to,
                  assume the defense of a claim shall not be obligated to pay
                  the fees and expenses of more than one counsel in any one
                  jurisdiction for all parties indemnified by such indemnifying
                  party with respect to such claim, unless in the reasonable
                  judgment of any indemnified party a conflict of interest may
                  exist between such indemnified party and any other of such
                  indemnified parties with respect to such claim, in which event
                  the indemnifying party shall be obligated to pay the fees and
                  expenses of such additional counsel or counsels.

9.7      Termination of Restrictions. Notwithstanding the foregoing provisions
         of this Section 9, the restrictions imposed by this Section 9 upon the
         transferability of the Warrants, the Warrant Stock and the Restricted
         Common Stock (or Common Stock issuable upon the exercise of the
         Warrants) and the legend requirement of Section 9.1 of this Agreement
         shall terminate as to any particular Warrant or share of Warrant Stock
         or Restricted Common Stock (or Warrant Stock) (i) when and so long as
         such security shall have been registered under the Securities Act and
         disposed of pursuant thereto, or (ii) when the Warrant Holder thereof
         shall have delivered to the Issuer the written opinion of counsel to
         such Warrant Holder, stating that such legend is not required in order
         to ensure


                                       24
<PAGE>   28

         compliance with the Securities Act. Whenever the restrictions imposed
         by this Section 9 shall terminate as to any Warrants or any Restricted
         Common Stock, as hereinabove provided, the Warrant Holder thereof shall
         be entitled to receive from the Issuer, at the expense of the Issuer, a
         new Warrant Certificate or a new certificate representing such Common
         Stock, as the case may be, not bearing the restrictive legend set forth
         in Section 9.1 of this Agreement.

9.8      Listing on Securities Exchange. If at any time the Issuer shall list
         any shares of Common Stock on any securities exchange, it will, at its
         expense, use its best efforts to list thereon, maintain and, when
         necessary, increase such listing of, all shares of Common Stock issued
         or, to the extent permissible under the applicable securities exchange
         rules, issuable upon the exercise of the Warrants so long as any shares
         of Common Stock shall be so listed during the Exercise Period.

10.      SUPPLYING INFORMATION

         The Issuer shall cooperate with each Warrant Holder of a Warrant and
         each Warrant Holder of Restricted Common Stock in supplying such
         information as may be reasonably necessary for such Warrant Holder to
         complete and file any information reporting forms presently or
         hereafter required by the Commission as a condition to the availability
         of an exemption from the Securities Act for the sale of any Warrant or
         Restricted Common Stock.

11.      LOSS OR MUTILATION

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

12.      OFFICE OF THE ISSUER

         As long as any of the Warrants remain outstanding, the Issuer shall
         maintain an office or agency (which may be the principal executive
         officers of the Issuer) where the Warrants may be presented for
         exercise, registration or transfer, division or combination as provided
         in this Agreement.

13.      APPRAISAL

         The determination of the Appraised Value per share of Common Stock
         shall be made by an investment banking firm of nationally recognized
         standing mutually agreed to by the Issuer and the Required Holders. If
         the investment banking firm selected by the Issuer is


                                       25
<PAGE>   29

         not acceptable to the Required Holders and the Issuer and the Required
         Holders cannot agree on a mutually acceptable investment banking firm,
         then the Required Holders and the Issuer shall each choose one such
         investment banking firm and the respective chosen firms shall agree on
         another investment banking firm which shall make the determination. The
         Issuer shall retain, at its sole cost, such investment banking firm as
         may be necessary for the determination of Appraised Value required by
         the terms of this Agreement.

14.      LIMITATION OF LIABILITY; NO RIGHTS AS STOCKHOLDER

         No provision hereof, in the absence of affirmative action by any
         Warrant Holder to purchase shares of Common Stock, and no enumeration
         herein of the rights or privileges of any Warrant Holder, shall give
         rise to any liability of such Warrant Holder for the purchase price of
         any Common Stock or as a stockholder of the Issuer, whether such
         liability is asserted by the Issuer or by creditors of the Issuer.
         Except as may otherwise be provided by law or by separate agreement
         between a Warrant Holder and the Issuer, no Warrant Holder, as such,
         shall be entitled to vote or be deemed the holder of Common Stock or
         any other securities (other than Warrants) of the Issuer which may at
         any time be issuable on the exercise hereof, nor shall anything
         contained herein be construed to confer upon any Warrant Holder the
         rights of a stockholder of the Issuer or the right to vote for the
         election of directors or upon any matters submitted to stockholders at
         any meeting thereof, or to give or withhold consent to any corporate
         action or to receive notice of meetings or other actions affecting
         stockholders (except as provided herein), or to receive dividends or
         otherwise, until the Warrants shall have been exercised in accordance
         with the terms and conditions hereof.

15.      MISCELLANEOUS

15.1     Non-waiver and Expenses. No course of dealing or any delay or failure
         to exercise any right hereunder on the part of any Warrant Holder shall
         operate as a waiver of such right or otherwise prejudice such Warrant
         Holder's rights, powers or remedies. If the Issuer fails to comply with
         any provision of this Agreement, the Issuer shall pay to the applicable
         Warrant Holders such amounts as shall be sufficient to cover any costs
         and expenses including, but not limited to, reasonable attorneys' fees,
         including those of appellate proceedings, incurred by the Warrant
         Holders in enforcing any of its rights, powers or remedies hereunder.

15.2     Notice Generally. Any notice, demand, request, consent, approval,
         declaration, delivery or other communication hereunder to be made
         pursuant to the provisions of this Agreement shall be sufficiently
         given or made if in writing and either delivered in person with receipt
         acknowledged or sent by registered or certified mail, return receipt
         requested, postage prepaid, telex, telecopier or overnight air courier
         guaranteeing next day delivery, addressed as follows:



                                       26
<PAGE>   30

         (a)      If to the Bank of Montreal, as Warrant Holder, at:

                           Address:       700 Louisiana, Ste. 4400
                                          Houston, Texas 77002
                           Attention:     Thomas E. McGraw
                           Telecopier No. (713) 223-0477

         (b)      If to the Issuer at:

                           Brigham Exploration Company
                           6300 Bridge Point Parkway
                           Building 2, Suite 500
                           Austin, Texas 78730
                           Attention: President
                           Telecopier No.: (512) 427-3300

                  or at such other address as may be substituted by notice given
                  as herein provided. The giving of any notice required
                  hereunder may be waived in writing by the party entitled to
                  receive such notice. Every notice, demand, request, consent,
                  approval, declaration, delivery or other communication
                  hereunder shall be deemed to have been duly given or served on
                  the date on which personally delivered, with receipt
                  acknowledged, or three (3) Business Days after the same shall
                  have been deposited in the United States mail.

15.3     Indemnification. Except to the extent otherwise provided in Section 9.6
         of this Agreement, the Issuer agrees to indemnify and hold harmless
         each Warrant Holder and its officers, directors, employees, agents,
         attorneys and Affiliates (each an "Indemnified Party") from and against
         any liabilities, obligations, losses, damages, penalties, actions,
         judgments, suits, claims, costs, attorneys' fees, expenses and
         disbursements of any kind which may be imposed upon, incurred by or
         asserted against such Indemnified Party relating to or arising out of
         (i) such Warrant Holder's exercise of the Warrants and/or ownership of
         any shares of Warrant Stock issued in consequence thereof, or (ii) any
         litigation to which such Warrant Holder is made a party in its capacity
         as a stockholder or Warrant Holder of the Issuer; provided, however,
         that the Issuer will not be liable hereunder to the extent that any
         liabilities, obligations, losses, damages, penalties, actions,
         judgments, suits, claims, costs, attorneys' fees, expenses or
         disbursements (A) arise solely from any violation by such Warrant
         Holder of any law or regulation applicable to it or (B) are found in a
         final non-appealable judgment by a court to have resulted from such
         Warrant Holder's bad faith or willful misconduct or violation of law.
         The procedures to be followed for claims of indemnification under this
         Section 15.3 shall be as set forth in Section 9.6(d) of this Agreement.

15.4     Remedies. Each Warrant Holder of Warrants and Warrant Stock, in
         addition to being entitled to exercise all rights granted by law,
         including recovery of damages, will be entitled to specific performance
         of its rights under Section 9 of this Agreement. The Issuer agrees that
         monetary damages would not be adequate compensation for any loss
         incurred by reason of a breach by it of the provisions of Section 9 of
         this Agreement, and


                                       27
<PAGE>   31

         hereby agrees to waive any defense to the contrary in any action for
         specific performance that a remedy at law would be adequate.

15.5     Successors and Assigns. Subject to the provisions of Sections 3.1 and 9
         of this Agreement, this Agreement and the rights evidenced hereby shall
         inure to the benefit of and be binding upon the successor of the Issuer
         and the successors and assigns of any Warrant Holder. The provisions of
         this Agreement are intended to be for the benefit of all Warrant
         Holders from time to time of the Warrants and Warrant Stock, and shall
         be enforceable by any such Warrant Holder.

15.6     Complete Agreement; Amendment. This Agreement and the Warrant
         Certificates constitute the complete agreement among the parties with
         respect to the subject matter hereof. This Agreement may be modified or
         amended or the provisions hereof waived only with the written consent
         of the Issuer and the Required Holders, provided that no Warrant may be
         modified or amended to reduce the number of shares of Common Stock for
         which such Warrant is exercisable or to increase the price at which
         such shares may be purchased upon exercise of such Warrant (before
         giving effect to any adjustment as provided herein) or to accelerate
         the Expiration Date without the prior written consent of the Warrant
         Holder thereof, and any amendment of Section 9 of this Agreement shall
         also require the written consent of Warrant Holders of Warrants and/or
         Warrant Stock representing more than 50% of the total of (i) all shares
         of Warrant Stock then subject to purchase upon exercise of all Warrants
         then Outstanding, and (ii) all shares of Warrant Stock then
         Outstanding.

15.7     Severability. Wherever possible, each provision of this Agreement shall
         be interpreted in such manner as to be effective and valid under
         applicable law, but if any provision of this Agreement shall be
         prohibited by or invalid under applicable law, such provision shall be
         ineffective to the extent of such prohibition or invalidity, without
         invalidating the remainder of such provision or the remaining
         provisions of this Agreement.

15.8     Headings. The headings used in this Agreement are for the convenience
         of reference only and shall not, for any purpose, be deemed a part of
         this Agreement.

15.9     Governing Law; Consent to Jurisdiction and Venue. In all respects,
         including all matters of construction, validity and performance, this
         Agreement and the obligations arising hereunder shall be governed by,
         and construed and enforced in accordance with, the laws of the State of
         Texas applicable to contracts made and performed in such state, without
         regard to the principles thereof regarding conflict of laws, and any
         applicable laws of the United States of America.

15.10    Consent to Jurisdiction and Venue.

         (a)      THE ISSUER AND EACH WARRANT HOLDER HEREBY EXPRESSLY SUBMITS TO
                  THE JURISDICTION OF THE COURTS OF THE STATE OF TEXAS OR THE
                  UNITED STATES DISTRICT COURT FOR THE SOUTHERN DISTRICT OF
                  TEXAS. FINAL JUDGMENT AGAINST SUCH PARTY IN ANY SUCH SUIT
                  SHALL BE CONCLUSIVE, AND MAY BE


                                       28
<PAGE>   32

                  ENFORCED IN ANY OTHER JURISDICTIONS BY SUIT ON THE JUDGMENT OR
                  AS OTHERWISE PERMITTED BY APPLICABLE LAW, A CERTIFIED OR TRUE
                  COPY OF WHICH SHALL BE CONCLUSIVE EVIDENCE OF THE FACTS AND OF
                  THE AMOUNT OF ANY INDEBTEDNESS OR LIABILITY OF SUCH PARTY
                  THEREIN DESCRIBED; PROVIDED, HOWEVER, EACH PARTY MAY AT ITS
                  OPTION BRING SUIT, OR INSTITUTE OTHER JUDICIAL PROCEEDINGS
                  AGAINST THE OTHER PARTY OR ANY OF ITS ASSETS, IN THE COURTS OF
                  ANY COUNTRY OR PLACE WHERE SUCH PARTY OR SUCH ASSETS MAY BE
                  FOUND.

         (b)      THE ISSUER AND EACH WARRANT HOLDER HEREBY IRREVOCABLY WAIVES
                  ANY OBJECTION WHICH IT MAY NOW OR HEREAFTER HAVE TO THE LAYING
                  OF VENUE OF ANY SUIT, ACTION OR PROCEEDING BROUGHT IN ANY
                  COURTS OF THE STATE OF TEXAS OR THE UNITED STATES DISTRICT
                  COURT FOR THE SOUTHERN DISTRICT OF TEXAS AND HEREBY FURTHER
                  IRREVOCABLY WAIVES ANY CLAIM THAT ANY SUCH SUIT, ACTION OR
                  PROCEEDING BROUGHT IN ANY SUCH COURT HAS BEEN BROUGHT IN AN
                  INCONVENIENT FORUM.

15.11    Counterparts: This Agreement may be executed in two or more
         counterparts, each of which shall be deemed an original, but all of
         which together shall constitute one and the same instrument.


                                       29
<PAGE>   33

         IN WITNESS WHEREOF, the parties hereto have caused this Agreement to be
duly executed as of the date first above written.



                                    BRIGHAM EXPLORATION COMPANY, as Issuer


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


                                    BANK OF MONTREAL, as Warrant Holder


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


                                       30
<PAGE>   34
                                                                    EXHIBIT A TO
                                                               Warrant Agreement



                          (FORM OF WARRANT CERTIFICATE)



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 THE WARRANT AGREEMENT, DATED AS OF JULY 19, 1999,
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.

No. ___

                               WARRANT CERTIFICATE

This Warrant Certificate certifies that the Bank of Montreal, a Canadian
chartered bank, or registered assigns thereof, is the holder of 661,538 warrants
(the "Warrants") to purchase shares of common stock of Brigham Exploration
Company, a Delaware corporation (the "Company"). Each Warrant entitles the
holder, but only subject to the conditions set forth herein and in the Warrant
Agreement referred to below, to purchase from the Company before 5:00 p.m., New
York City time, on the Expiration Date, as such term is defined in the Warrant
Agreement, one fully paid and non-assessable share of common stock of the
Company ("Warrant Stock") at a price (the "Exercise Price") of two and 25/100
dollars ($2.25) per share of Warrant Stock payable as set forth in the Warrant
Agreement. The number of shares of Warrant Stock for which each Warrant is
exercisable and the Exercise Price are each subject to adjustment prior to the
Expiration Date upon the occurrence of certain events as set forth in the
Warrant Agreement.

The Company may deem and treat the registered holders of the Warrants evidenced
hereby as the absolute owner thereof (notwithstanding any notation of ownership
or other writing hereon made by anyone), for the purpose of any exercise hereof
and of any distribution to the holders hereof, and for all other purposes.

Warrant Certificates, when surrendered at the office of the Company by the
registered holder hereof in person or by a legal representative duly authorized
in writing, may be exchanged, in the


                                    EXH. A-1
<PAGE>   35
manner and subject to the limitations provided in the Warrant Agreement, but
without payment of any service charge, for another Warrant Certificate or
Warrant Certificates of like tenor evidencing in the aggregate a like number of
Warrants.

Upon due presentment for registration of transfer of this Warrant Certificate at
the office of the Company at 6300 Bridge Point Parkway, Building 2, Suite 500,
Austin, Texas 78730, Attention: President, or such other address as the Company
may specify in writing to the registered holder of the Warrants evidenced
hereby, a new Warrant Certificate or Warrant Certificates of like tenor and
evidencing in the aggregate a like number of Warrants shall be issued to the
transferee in exchange for this Warrant Certificate to the transferee(s) and, if
less than all the Warrants evidenced hereby are to be transferred, the
registered holder hereof, subject to the limitations provided in the Warrant
Agreement, without charge except for any tax or other governmental charge
imposed in connection therewith.

The Warrant Certificate is one of the Warrant Certificates referred to in the
Warrant Agreement, dated as of July 19, 1999, between the Company and the
initial holder of Warrants party thereto (the "Warrant Agreement"). Said Warrant
Agreement is hereby incorporated by reference in and made a part of this
instrument and is hereby referred to for a description of the rights, limitation
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 Warrant Agreement, the provisions of the
Warrant Agreement shall control.

IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to be duly
executed and its corporate seal to be impressed hereon and attached by its
secretary.

Dated: ____________________________, 1999.


                                    BRIGHAM EXPLORATION COMPANY


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


(CORPORATE SEAL)

ATTEST:


--------------------------------------
Secretary


                                    EXH. A-2
<PAGE>   36
                                                                    Exhibit A To
                                                             Warrant Certificate


                                SUBSCRIPTION FORM

                 [To be executed only upon exercise of Warrants]


The undersigned registered owner of this Warrant Certificate irrevocably
exercises Warrants for the purchase of shares of Common Stock of [     ] and
herewith makes payment therefor

         $_____   in cash

all at the price and on the terms and conditions specified in the Warrant
Certificate and the Warrant Agreement, and requests that certificates for the
shares of Common Stock hereby purchased (and any securities or other property
issuable upon such exercise) be issued in the name of ____________________ and
delivered to _____________________________ whose address is ___________________
________________________________ and, if such shares of Common Stock shall not
include all of the shares of Common Stock issuable as provided in the Warrant
Certificate, that a new Warrant Certificate of like tenor and date for the
balance of the shares of Common Stock issuable thereunder be delivered to the
undersigned.



----------------------------------------------
(Name of Registered Owner)



----------------------------------------------
(Signature of Registered Owner)



----------------------------------------------
(Street Address)



----------------------------------------------
(City)            (State)           (Zip Code)


                                    EXH. A-3
<PAGE>   37
                                                                    Exhibit B To
                                                             Warrant Certificate


                                 ASSIGNMENT FORM


FOR VALUE RECEIVED the undersigned registered owner of the attached Warrant
Certificate hereby sells, assigns and transfers unto the assignee named below
all of the rights of the undersigned under this Warrant Certificate, with
respect to the number of shares of Common Stock set forth below:

Name and Address of Assignee:
                             ----------------------------

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

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

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

No. of Shares of
Common Stock ______

and does hereby irrevocably constitute and appoint ____________________________
attorney-in-fact to register such transfer on the books of [    ] maintained for
that purpose, with full power of substitution in the premises.

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

Name:
     --------------------------------

Signature:
          ---------------------------

Witness:
        -----------------------------

The assignee named above hereby agrees to purchase and take the attached Warrant
Certificate pursuant to and in accordance with the terms and conditions of the
Warrant Agreement, dated as of ______________, 1999, between [    ] and the
initial Holder named therein and agrees to be bound thereby.

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

Name:
     --------------------------------

Signature:
          ---------------------------


                         EXH. B-1 to Warrant Certificate
<PAGE>   38
                                                                    EXHIBIT B TO
                                                               Warrant Agreement



                      [FORM OF CASHLESS CONVERSION NOTICE]

             (To be executed upon a cashless exercise of a Warrant.)

         The undersigned hereby irrevocably elects to exercise the Cashless
Conversion, represented by this Warrant Certificate, to purchase ____ shares of
Common Stock and herewith tenders in payment for such shares this Warrant
Certificate, all in accordance with the terms hereof. The undersigned requests
that a certificate for such shares be registered in the name of _______________
_______________________________________________ whose address is ______________
________________________________________________ and that such certificate (or
any payment in lieu thereof) be delivered to ______________________________
whose address is ___________________________________.


Dated:
      -----------------------            --------------------------------------
                                         (Signature must conform in all respects
                                         to name of holder as specified on the
                                         face of the Warrant.)


         And if said number of shares shall not be all the shares exchangeable
or purchasable under the within Warrant, a new Warrant of like tenor is to be
issued in the name of the undersigned for the balance remaining of the shares
purchasable thereunder.


                                    EXH. B-1
<PAGE>   39
                                   SCHEDULE A


<TABLE>
<CAPTION>
                          Shares of                   Stock Options
                           Common                      Outstanding
                            Stock         -----------------------------------------         Warrants
                         Outstanding      Vested         Unvested           Total         Outstanding
                         -----------      -------        ---------        ---------       -----------
<S>                      <C>              <C>            <C>              <C>             <C>
As of July 19, 1999      14,309,071       267,104        1,307,383        1,574,487        1,000,000

     Issued 7/20/99         119,550
     Forfeited 7/30/99                                     (10,500)         (10,500)
     Forfeited 8/16/99                                    (263,888)        (263,888)
     Issued 8/17/99          89,165                        105,000          105,000
                         ----------       -------        ---------        ---------        ---------
As of August 18, 1999*   14,517,786       267,104        1,137,995        1,405,099        1,000,000
                         ----------       -------        ---------        ---------        ---------
</TABLE>


                                    SCH. A-1