Document:

REGISTRATION RIGHTS AGREEMENT

         THIS REGISTRATION RIGHTS AGREEMENT (this "AGREEMENT") is entered into
as of November 1, 2000, by and among Brigham Exploration Company, a Delaware
corporation (the "COMPANY"), and the parties listed on SCHEDULE A hereto (the
"INVESTORS").

                                    RECITALS

                  WHEREAS, each Investor is a party to a certain Warrant
Certificate (the "WARRANT CERTIFICATE") dated as of November 1, 2000, by and
between the Company and such Investor;

                  WHEREAS, the Warrant Certificates were executed and delivered
in connection with the consummation of transactions contemplated by that certain
Securities Purchase Agreement dated as of November 1, 2000 by and among the
Company and the Investors;

                  WHEREAS, pursuant to the Warrant Certificates, the Holder (as
defined below) has been issued a warrant (the "WARRANT") to purchase shares of
the Company's common stock, par value $.01 per share (the "COMPANY'S COMMON
STOCK");

                  WHEREAS, to induce the Investors to enter into the Warrant
Certificates and the Securities Purchase Agreement, the Company has agreed to
grant the registration and other rights contained in this Agreement;

                                    AGREEMENT

         1.       DEFINITIONS

         For purposes of this Agreement, the following terms have the following
meanings:

         (1) "FORM S-3" means such form under the Act as in effect on the date
hereof or any registration form under the Act subsequently adopted by the
Securities and Exchange Commission (the"SEC") that similarly permits inclusion
or incorporation of substantial information by reference to other documents
filed by the Company with the SEC;

         (2) "HOLDER" means any person owning or having the right to acquire
Registrable Securities who is a party to this Agreement as of the date hereof or
who may be added as a party pursuant to the terms of this Agreement, and any
assignee thereof;

         (3) "REGISTER," "REGISTERED" and "REGISTRATION" refer to a registration
effected by preparing and filing a registration statement or similar document in
compliance with the Securities Act of 1933, as amended (the "ACT"), and the
declaration or order of effectiveness of such registration statement or
document;

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<PAGE>

         (4) "REGISTRABLE SECURITIES" means (i) the Common Stock of the Company
issuable or issued upon exercise of the Warrants and (ii) any Common Stock of
the Company issued as (or issuable upon the conversion or exercise of any
warrant, right or other security which is issued as) a dividend or other
distribution with respect to, or in exchange for or in replacement of, such
Warrants or Common Stock, excluding in all cases, however, any Registrable
Securities sold by a person in a transaction in which its rights under this
Agreement are not assigned ; and

         (5) "REGISTRABLE SECURITIES THEN OUTSTANDING" means the number of
shares of Common Stock outstanding which are, and the number of shares of Common
Stock issuable pursuant to then exercisable or convertible securities which are,
Registrable Securities.

         (6) "SEC" means the Securities and Exchange Commission.

         2. REQUEST FOR REGISTRATION

         (1) If the Company shall receive a written request from the Holders of
at least 25% of the Registrable Securities then outstanding (the "INITIATING
HOLDERS") that the Company file a registration statement under the Act covering
the registration of at least 25% of the Registrable Securities, then the Company
shall, within 10 days after the receipt of such request, give written notice of
such request to all Holders and shall, subject to the limitations set forth
below, use its reasonable best efforts to effect as soon as practicable the
registration under the Act of all Registrable Securities that the Holders
request to be registered in a written request to be given within 30 days of
receipt of such notice by the Company.

         (2) The Company is obligated to effect only two registrations pursuant
to this Section 2.

         (3) Notwithstanding the foregoing, if the Company shall furnish to the
Initiating Holders requesting a registration pursuant to this Section 2 within
30 days of receiving such request a certificate signed by the President of the
Company stating that in the good faith Judgment of the Board of Directors of the
Company it would be seriously detrimental to the Company and its stockholders
for such in registration statement to be filed and it is therefore essential to
defer the filing of such registration statement, the Company shall have the
right to defer such filing for up to 2 periods of not more than 45 days each
after receipt of the request of the Initiating Holders; provided, however, that
the Company may not use this right more than once (for a total of up to 90 days)
in any 12-month period; provided, however, that the Company shall promptly
notify the Initiating Holders requesting a registration pursuant to this Section
2 of any decision by the Company to abandon or indefinitely delay such public
offering.

         3. COMPANY REGISTRATION

         If the Company proposes to register (including for this purpose a
registration effected by the Company for stockholders (ARequesting Holders@)
other than the Holders) any of its stock or other securities under the Act in
connection with the public offering of such securities solely for

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<PAGE>

cash (other than a registration relating solely to the sale of securities to
participants in a Company stock plan, or a registration on any form that does
not include substantially the same information as would be required to be
included in a registration statement covering the sale of Registrable
Securities), the Company shall, at each such time, promptly give each Holder
written notice of such registration. Upon the written request of each such
Holder given within 20 days after receipt of such notice by the Company, the
Company shall, subject to the provisions of Section 8, use its reasonable best
efforts to cause to be registered under the Act all of the Registrable
Securities that each such Holder has requested to be registered. In the event
that the Company decides for any reason not to complete the registration of
shares of Common Stock other than Registrable Securities, the Company shall have
no obligation under this Section 3 to continue with the registration of
Registrable Securities. Any request pursuant to this Section 3 to register
Registrable Securities as part of an underwritten public offering of Common
Stock shall specify that such Registrable Securities are to be included in the
underwriting on the same terms and conditions as the shares of Common Stock
otherwise being sold through underwriters under such registration.

         4. FORM S-3 REGISTRATION

         (1) If the Company shall receive a written request from the Holders of
25% of the Registrable Securities then outstanding that the Company effect a
registration on Form S-3, the reasonably anticipated aggregate offering price to
the public of which would equal or exceed $2,000,000,then the Company shall,
within 10 days after the receipt of such request, give written notice of such
request to all Holders and shall, subject to the limitations set forth below,
use its reasonable best efforts to effect as soon as practicable the
registration under the Act of all Registrable Securities that the Holders
request to be registered in a written request to be given within 30 days of
receipt of such notice by the Company.

         (2) Notwithstanding the foregoing, the Company shall not be obligated
to effect any such registration pursuant to this Section 4 if (i) Form S-3 is
not available for such offering by the Holders; (ii) the Company shall furnish
to the Holders a certificate signed by the President of the Company stating
that, in the good faith judgment of the Board of Directors of the Company, it
would be seriously detrimental to the Company and its stockholders for such Form
S-3 registration to be effected at such time, in which event the Company shall
have the right to defer the filing of the Form S-3 registration statement for a
period of not more than 90 days after receipt of the request of the Holders
under this Section 4; provided, however, that the Company shall not use this
right more than once in any 12-month period; or (iii) if the Company has, within
the 12-month period preceding the date of such request, already effected one
such registrations on Form S-3 for the Holders pursuant to this Section 4.

         5. OBLIGATIONS OF THE COMPANY

         Whenever required under this Agreement to effect the registration of
any Registrable Securities, the Company shall, as expeditiously as reasonably
possible:

         (1) Prepare and file with the SEC a registration statement with respect
to such Registrable Securities and use its commercially reasonable efforts to
cause such registration statement to become effective, and, upon the request of
the Holders of 25% of the

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Registrable Securities registered thereunder, keep such registration statement
effective for up to one year.

         (2) Prepare and file with the SEC such amendments and supplements to
such registration statement and the prospectus used in connection with such
registration statement as may be necessary to comply with the provisions of the
Act with respect to the disposition of all securities covered by such
registration statement.

         (3) Furnish to the Holders such copies of a prospectus, including a
preliminary prospectus, in conformity with the requirements of the Act and such
other documents as they may reasonably request to facilitate the disposition of
all securities covered by such registration statement

         (4) Use commercially reasonable efforts to register and qualify the
securities covered by such registration statement under such other securities or
blue sky laws of such jurisdictions as shall be reasonably requested by the
Holders, provided that the Company shall not be required to qualify to do
business or to file a general consent to service of process in any such states
or jurisdictions.

         (5) In the event of any underwritten public offering, enter into and
perform its obligations under an underwriting agreement, in usual and customary
form, with the managing underwriter(s) of such offering. Each Holder
participating in such registration shall also enter into and perform its
obligations under such an agreement.

         (6) Notify each Holder of Registrable Securities covered by such
registration statement, during the time when a prospectus is required to be
delivered under the Act, of the happening of any event as a result of which the
prospectus included in such registration statement, as then in effect, includes
an untrue statement of a material fact or omits to state a material fact
required to be stated therein or necessary to make the statements therein not
misleading in the light of the circumstances then existing.

         (7) At the request of any Holder selling Registrable Securities in such
registration, furnish on the date that such Registrable Securities are delivered
to the underwriters for sale in connection with such registration (i) an
opinion, dated such date, of legal counsel representing the Company for the
purposes of such registration, in form and substance as is customarily given by
Company counsel to underwriters in an underwritten public offering, addressed to
the underwriters and (ii) a letter, dated such date, from the independent
certified public accountants of the Company, in form and substance as is
customarily given by independent certified public accountants to underwriters in
an underwritten public offering, addressed to the underwriters.

         (8) List the Registrable Securities being registered on any national
securities exchange on which a class of the Company's equity securities is
listed or qualify the Registrable Securities being registered for inclusion on
Nasdaq if the Company does not have a class of equity securities listed on a
national securities exchange.

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<PAGE>

         6. FURNISH INFORMATION

         It shall be a condition precedent to the obligations of the Company to
take any action pursuant to this Agreement that the selling Holders shall
furnish to the Company such information regarding themselves, the Registrable
Securities held by them and the intended method of disposition of such
securities as shall be reasonably required to effect the registration of their
Registrable Securities and shall execute such documents in connection with such
registration as the Company may reasonably request.

         7. EXPENSES OF REGISTRATION

         In connection with any registration pursuant to this Agreement, the
Company shall be responsible for the payment of all reasonable expenses of the
registration, with the exception of (i) underwriting discounts and commissions,
which shall be paid by the Company, the Holders and any other selling holders of
the Company's securities in proportion to the aggregate value of the securities
offered for sale by each of them, and (ii) the fees and expenses of more than
one law firm acting as counsel to the selling Holders selected by a majority in
interest of the selling Holders, which additional counsel, if any, shall be paid
by the Holder or Holders that engage such counsel. The expenses to be paid by
the Company shall include, without limitation, all registration, filing and
qualification fees, printing and accounting fees, the fees and disbursements of
counsel for the Company and the fees and disbursements of one counsel for the
selling Holders; provided, however, that the Company shall not be required to
pay for any expenses of any registration proceeding begun pursuant to Section 2
or 4 if the registration request is subsequently withdrawn (other than a
withdrawal due to a material adverse change in the Company's business or
financial condition), unless, in the event of a registration initiated pursuant
to the provisions of Section 2, the Holders of 25% of the Registrable Securities
agree to forfeit the right to one demand registration.

         8. UNDERWRITING REQUIREMENTS

         (1) The Holders requesting registration under Section 2 must distribute
the Registrable Securities covered by their request by means of a public
offering underwritten by a reputable national or regional underwriter. The right
of any Holder to include its Registrable Securities in such registration under
Section 2 shall be conditioned upon such Holder's participation in such
underwriting and the inclusion of such Holder's Registrable Securities in the
underwriting to the extent provided herein. All Holders proposing to distribute
their Registrable Securities through such underwriting shall (together with the
Company as provided in Section 5(e)) enter into an underwriting agreement in
customary form with the underwriter or underwriters selected for such
underwriting by a majority in interest of the Initiating Holders.
Notwithstanding any other provision of Section 2, if the underwriter advises the
Initiating Holders in writing that marketing factors require a limitation of the
number of shares to be underwritten, then the Initiating Holders shall so advise
all Holders of Registrable Securities which would otherwise have been
underwritten pursuant to Section 2, and the number of shares of Registrable
Securities that may be included in the registration shall be apportioned first
pro rata among the selling Holders, including the Initiating Holders, according
to the total amount of Registrable Securities requested to be sold in such
registration by such Holders, then to the Company and then pro rata among any
other selling stockholders according to the total amount of securities otherwise
entitled to be included therein

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<PAGE>

owned by each such selling stockholder, or in such other proportions as shall
mutually be agreed to by such selling stockholders.

         (2) The Company shall not be required under Section 3 to include any of
the Holders' securities in an underwritten offering of the Company's securities
unless such Holders accept the terms of the underwriting as agreed upon between
the Company and the underwriters selected by it. If the underwriters advise the
Company that marketing factors require a limitation on the number of shares,
including Registrable Securities, to be included in such offering, then the
Company shall so advise all Holders of Registrable Securities that would
otherwise have been underwritten pursuant to Section 3, and the number of
shares, including Registrable Securities, that may be included in the
registration shall be apportioned first to the Company or Requesting Holders, as
the case may be, then pro rata among the selling Holders according to the total
amount of Registrable Securities requested to be sold in such registration by
such Holders, then pro rata among any other selling stockholders according to
the total amount of securities otherwise entitled to be included therein owned
by each such other selling stockholder, or in such other proportions as shall
mutually be agreed to by such selling stockholders; provided that in no event
shall the amount of securities of the selling Holders included in the
registration be reduced below 20% of the total amount of securities included in
such registration.

         9. DELAY OF REGISTRATION

         No Holder shall have any right to obtain or seek an injunction
restraining or otherwise delaying any registration of the Company as the result
of any controversy that might arise with respect to the interpretation or
implementation of this Agreement.

         10. INDEMNIFICATION

         In the event any Registrable Securities are included in a registration
statement under this Agreement:

         (1) To the extent permitted by law, the Company will indemnify and hold
harmless each Holder, any underwriter (as defined in the Act) for such Holder
and each person, if any, who controls such Holder or underwriter within the
meaning of the Act or the Securities Exchange Act of 1934, as amended (the "1934
ACT"), against any actual expenses (including legal fees and costs), losses,
claims, damages (including settlement amounts) or liabilities joint or several)
(collectively, "LOSSES") to which they may become subject under the Act, the
1934 Act or other federal or state law, insofar as such Losses arise out of or
are based upon any of the following statements, omissions or violations
(collectively, a "VIOLATION"): (i) any untrue statement or alleged untrue
statement of a material fact contained in such registration statement, including
any preliminary prospectus or final prospectus contained therein, or any
amendments or supplements thereto, untrue in light of the circumstances under
which they were made, (ii) the omission or 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, or (iii) any violation or alleged violation by the Company of the
Act, the 1934 Act, any state securities law or any rule or regulation
promulgated under the Act, the 1934 Act or any state securities law. The Company
will reimburse (as incurred) each such Holder, underwriter or controlling person
for

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any Losses reasonably incurred by them in connection with investigating or
defending any Violations; provided, however, that the indemnity agreement
contained in this Section 10(a) shall not apply to amounts paid in settlement of
any claims for Violations if such settlement is made without the consent of the
Company, which consent shall not be unreasonably withheld, nor shall the Company
be liable in any such case for any Losses that arise out of or are based upon a
Violation that occurs in reliance upon and in conformity with written
information furnished expressly for use in connection with such registration by,
or on behalf of, any such Holder, underwriter or controlling person.

         (2) To the extent permitted by law, each selling Holder will indemnify
and hold harmless the Company and its officers, directors, agents and employees,
each underwriter and each other Holder selling securities in such in
registration statement, and any person who controls any of the foregoing within
the meaning of the Act or the 1934 Act, against any Losses to which the Company
or such officer, director, agent, employee, or underwriter or other selling
Holder or controlling person may become subject under the Act, the 1934 Act or
other federal or state law, insofar as such Losses arise out of or are based
upon any Violation that occurs in reliance upon and in conformity with written
information furnished by, or on behalf of, such Holder expressly for use in
connection with such registration; and each such Holder will reimburse any
Losses reasonably incurred by the Company or its officers, directors, agents,
employees, or underwriters or other selling Holders or controlling persons in
connection with investigating or defending any Violations; provided, however,
that (i) the indemnity agreement contained in this Section 10(b) shall not apply
to amounts paid in settlement of any claims for Violations if such settlement is
made without the consent of the Holder, which consent shall not be unreasonably
withheld and (ii) the obligations of such Holders shall be limited to an amount
equal to the gross proceeds before expenses and commissions to each such Holder
of Registrable Securities sold as contemplated herein.

         (3) Promptly after receipt of notice of the commencement of any action
(including any governmental action), an indemnified party will, if a claim is to
be made against any indemnifying party under this Section 10, deliver to the
indemnifying party a written notice of the commencement, and the indemnifying
party shall have the right to participate in, and, to the extent the
indemnifying party so desires, jointly with any other indemnifying party
similarly notified to assume the defense thereof with counsel mutually
satisfactory to the parties; provided, however, that an indemnified party shall
have the right to retain its own counsel, with the fees and expenses to be paid
by the indemnifying Party, if, in the opinion of counsel for the indemnifying
party, representation of such indemnified party by the counsel retained by the
indemnifying party would be inappropriate due to actual or potential differing
interests between such indemnified party and any other party represented by such
counsel in the proceeding. The failure to deliver written notice to the
indemnifying party within a reasonable period of time after notice of the
commencement of any such action shall relieve such indemnifying party of any
liability to the indemnified party under this Section 10 to the extent such
failure is prejudicial to its ability to defend such action, but the omission to
deliver written notice to the indemnifying party will not relieve it of any
liability that it may have to any indemnified party otherwise than under this
Section 10.

         (4) If the indemnification provided for in this Section 10 is held by a
court of competent jurisdiction to be unavailable to an indemnified party with
respect to any

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Losses, 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 in such proportion as is appropriate to reflect the
relative fault of the indemnifying party on the one hand and of the indemnified
party on the other in connection with the Violations that resulted in such
Losses as well as any other relevant equitable considerations; provided, that,
in no event shall any contribution by a Holder under this Section 10(d) exceed
the gross proceeds before expenses and commissions to each such Holder, except
in the case of willful fraud by such Holder. The relative fault of the
indemnifying party and of the indemnified party shall be determined by reference
to, among other things, whether the Violation resulting in such Losses relates
to information supplied by the indemnifying party or by the indemnifying party
and the parties= relative intent, knowledge, access to information, and
opportunity to correct or prevent such Violation.

         (5) Notwithstanding the foregoing, to the extent that the provisions on
indemnification and contribution contained in the underwriting agreement entered
into in connection with the underwritten public offering are in conflict with
the foregoing provisions, the provisions in the underwriting agreement shall
control.

         (6) The obligations of the Company and Holders under this Section 10
shall survive the completion of any offering of Registrable Securities and the
termination of Registration Rights pursuant to Section 15.

         11. REPORTS UNDER THE ACT

         With a view to making available to the Holders the benefits of SEC Rule
144 promulgated under the Act and any other rule or regulation of the SEC that
may at an time permit a Holder to sell securities of the Company to the public
without registration or pursuant to a registration on Form S-3, the Company
agrees to use commercially reasonable efforts to:

         (1) Make and keep public information available, as those terms are
understood and defined in SEC Rule 144, at all times after the effective date of
the first registration statement filed by the Company for the offering of its
securities to the general public;

         (2) File with the SEC in a timely manner all reports and other
documents required of the Company under the Act and the 1934 Act; and

         (3) Furnish to any Holder, so long as the Holder owns any Registrable
Securities, promptly upon request (i) a written statement by the Company that it
has complied with the reporting requirements of the 1934 Act (at any time after
the date on which it becomes subject to such reporting requirements), or that it
qualifies as a registrant whose securities may be resold pursuant to Form S-3
(at any time after it so qualifies), (ii) a copy of the most recent annual or
quarterly report of the Company and such other reports and documents so filed by
the Company, and (iii) such other information as may be reasonably requested in
availing any Holder of any rule or regulation of the SEC that permits the
selling of any such securities without registration or pursuant to such Form
S-3.

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         12. ASSIGNMENT OF REGISTRATION RIGHTS

         The rights to cause the Company to register Registrable Securities
pursuant to this Agreement may be assigned by a Holder to a transferee or
assignee of such securities who shall, upon such transfer or assignment, be
deemed a AHolder@ under this Agreement; provided that the Company is, within a
reasonable period of time after such transfer, furnished with written notice of
the name and address of such transferee or assignee and the securities with
respect to which such registration rights are being assigned; provided, further,
that such assignment shall be effective only if immediately following such
transfer the further disposition of such securities by the transferee or
assignee is restricted under the Act and that such transferee or assignee is (a)
a partner or retired partner of any Holder that is a partnership, (b) a member
of the immediate family or a trust for the benefit of any Holder that is an
individual, (c) an entity controlling, controlled by or under common control
with any Holder that is not an individual, (d) a transferee or assignee that
after the transfer or assignment holds (i) 10% of the Registrable Securities of
the transferor, or (ii) Registrable Securities prior to the transfer, or (e) a
constituent member of any Holder that is a limited liability company.

         13. LIMITATIONS ON SUBSEQUENT REGISTRATION RIGHTS

         From and after the date of this Agreement, the Company shall not,
without the prior written consent of the Holders of 25% of the Registrable
Securities then outstanding, enter into any agreement with any holder or
prospective holder of any securities of the Company that would allow such holder
or prospective holder to (a) include such securities in any registration filed
under Section 2, 3 or 4, unless under the terms of such agreement such holder or
prospective holder may include such securities in any such registration only to
the extent that the inclusion of its securities will not reduce the amount of
the Registrable Securities of the Holders which is included or (b) make a demand
registration which could result in such registration statement being declared
effective prior to 180 days of the effective date of any registration effected
pursuant to Section 2.

         14. "MARKET STANDOFF" AGREEMENT

         The Holders hereby agree that they shall not, to the extent requested
by the Company and an underwriter of Common Stock (or other securities) of the
Company, sell or otherwise transfer or dispose (other than to donees who agree
to be similarly bound) of any Registrable Securities for 180 days following the
effective date of a registration statement of the Company filed under the Act;
provided, however, that the foregoing shall not be effective unless all officers
and directors of the Company (whether or not pursuant to this Agreement) enter
into similar agreements and the Company has used all reasonable efforts to
obtain similar agreements from all holders of at least 1% of the Company=s then
outstanding Common Stock.

         To enforce the foregoing covenant, the Company may impose stop-transfer
instructions with respect to the Registrable Securities of the Holders (and the
shares or securities of every other person subject to the foregoing restriction)
until the end of such period.

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<PAGE>

         15. TERMINATION OF REGISTRATION RIGHTS

         The registration rights granted under Sections 2, 3 and 4 of this
Agreement shall terminate as to each Holders on the earlier of (a) the tenth
(10th) anniversary of the date of this Agreement and (b) the date on which all
Registrable Securities held by or issuable to such Holder (and its affiliates)
may be sold under Rule 144 during any 90 day period.

         15. MISCELLANEOUS

         15.1 NOTICES

         Any notice required or permitted under this Agreement shall be given in
writing and shall be deemed effectively given (a) upon personal delivery to the
party to be notified, (b) upon confirmation of receipt by fax by the party to be
notified, (c) one business day after deposit with a reputable overnight courier,
prepaid for overnight delivery and addressed as set forth in (d), or (d) three
days after deposit with the United States Post Office, postage prepaid,
registered or certified with return receipt requested and addressed to the party
to be notified at the address indicated for such party on the signature page, or
at such other address as such party may designate by 10 days= advance written
notice to the other parties given in the foregoing manner.

         15.2 AMENDMENTS AND WAIVERS

         Any term of this Agreement may be amended and the observance of any
term may be waived (either generally or in a particular instance and either
retroactively or prospectively) only with the written consent of the Company and
the holders of a majority of the Registrable Securities then outstanding.
Additional Holders may be added to this Agreement with such consent by amending
SCHEDULE A and adding a signature page executed by such additional Holder.

         15.3 GOVERNING LAW; JURISDICTION; VENUE

         This Agreement shall be governed by and construed under the laws of the
State of New York without regard to principles of conflict of laws. The parties
irrevocably consent to the jurisdiction and venue of the state and federal
courts located in New York City in connection with any action relating to this
Agreement.

         15.4 SUCCESSORS AND ASSIGNS

         The terms and conditions of this Agreement shall inure to the benefit
of and be binding on the respective successors and assigns of the parties as
provided herein.

         15.5. SEVERABILITY

         If one or more provisions of this Agreement are held to
beunenforceable under applicable law, such provision shall be excluded from
this Agreement, and the balance of this Agreement shall be interpreted as if
such provision were so excluded and shall be enforceable in accordance with its
terms.

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<PAGE>

         15.6 ENTIRE AGREEMENT; COUNTERPARTS

         This Agreement constitutes the entire agreement between the parties
about its subject and supersedes all prior agreements. This Agreement may be
executed in two or more counterparts, which together shall constitute one
instrument.

                            [Signature page follows]

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<PAGE>

         IN WITNESS WHEREOF, the parties have executed this Agreement as of the
date first above written.

                                       BRIGHAM EXPLORATION COMPANY

                                       By: /S/ KAREN E. LYNCH
                                          -------------------------------------
                                                Its: Vice President

                                       Address:

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

                                       DLJ MB FUNDING III, INC.

                                       By: /S/ IVY B. DODES
                                          -------------------------------------
                                               Ivy B. Dodes
                                               Principal

                                       Address:

                                       277 Park Avenue
                                       New York, New York 10172
                                       Attention: Michael Isikow
                                       Telefax: 212-892-2689

                                       DLJ ESC II, LP

                                       By:      DLJ LBO Plans Management
                                                Corporation, its general partner

                                                By: /S/ IVY B. DODES
                                                   ----------------------------
                                                        Ivy B. Dodes
                                                        Principal
                                       Address:

                                       277 Park Avenue
                                       New York, New York 10172

                                       Attention: Michael Isikow
                                       Telefax: 212-892-2689

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<PAGE>

                                   SCHEDULE A
                        TO REGISTRATION RIGHTS AGREEMENT

 HOLDER NAME                                     NUMBER OF SHARES
 -----------------------                         ----------------

 DLJ MB Funding III, Inc.,                           6,036,667
 a Delaware corporation

 DLJ ESC II, LP,                                       630,000
 a Delaware limited partnership

                                       13THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN REGISTERED UNDER
THE SECURITIES ACT OF 1933, AS AMENDED, OR ANY APPLICABLE STATE SECURITIES LAWS,
AND MAY NOT BE SOLD OR TRANSFERRED IN THE ABSENCE OF SUCH REGISTRATION OR AN
EXEMPTION THEREFROM. SUCH SECURITIES ARE SUBJECT TO THE RESTRICTIONS AND
PRIVILEGES SPECIFIED IN THIS WARRANT CERTIFICATE AND IN A SECURITIES PURCHASE
AGREEMENT, DATED AS OF NOVEMBER 1, 2000, BETWEEN BRIGHAM EXPLORATION COMPANY AND
THE INITIAL HOLDER OF SECURITIES NAMED THEREIN, A COPY OF WHICH IS ON FILE WITH
THE SECRETARY OF BRIGHAM EXPLORATION COMPANY AND WILL BE FURNISHED WITHOUT
CHARGE TO THE HOLDER HEREOF UPON WRITTEN REQUEST, AND THE HOLDER OF THIS
CERTIFICATE AGREES TO BE BOUND THEREBY.

                               WARRANT CERTIFICATE

NUMBER OF WARRANTS: 6,036,667                                    WARRANT NO.  1

         This Warrant certificate ("Warrant Certificate") certifies that, for
value received, DLJ MB Funding III, Inc., a Delaware corporation, is the
registered holder of the number of warrants (the "Warrants") set forth above.
Each Warrant entitles the holder thereof, at any time or from time to time
during the Exercise Period, to purchase from the Company one fully paid and
nonassessable share of Common Stock at the Exercise Price, subject to adjustment
as provided herein. Initially capitalized terms used but not defined herein
shall have the meanings ascribed to them in the Securities Purchase Agreement.

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

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

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

                  "EXERCISE PRICE," subject in all circumstances to adjustment
         in accordance with SECTION 2, means $3.00 per share.

                  "EXPIRATION DATE" means the tenth anniversary of the Closing
         Date.

                  "ISSUANCE DATE" means November 1, 2000.

                                       1
<PAGE>

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

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

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

                  "PRICE" means the average of the "high" and "low" prices as
         reported in THE WALL STREET JOURNAL'S listing for such day (corrected
         for obvious typographical errors) or if such shares are not reported in
         such listing, the average of the reported sales prices on the largest
         national securities exchange (based on the aggregate dollar value of
         securities listed) on which such shares are listed or traded, or if
         such shares are not listed or traded on any national securities
         exchange, then the average of the reported sales prices for such shares
         in the over-the-counter market, as reported on the National Association
         of Securities Dealers Automated Quotations System, or, if such prices
         shall not be reported thereon, the average of the closing bid and asked
         prices so reported, or, if such prices shall not be reported, then the
         average of the closing bid and asked prices reported by the National
         Quotations Bureau Incorporated. The "Average" Price per share for any
         period shall be determined by dividing the sum of the Prices determined
         for the individual trading days in such period by the number of trading
         days in such period.

                  "SECURITIES PURCHASE AGREEMENT" means the Securities Purchase
         Agreement, dated as of November 1, 2000, between the Company, DLJ MB
         Funding III, Inc. and DLJ ESC II, LP.

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

         Section 1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in
whole or in part, at any time or from time to time, during the Exercise Period,
by (i) presentation and surrender to the Company at its address set forth in
SECTION 10 of this Warrant Certificate with the Election To Exercise, attached
hereto as EXHIBIT A, duly completed and executed, and (ii) payment of the
Exercise Price, for the number of Warrants being exercised by either: (1) bank
draft or cashiers check, or (2) provided that the Company receives at least 5
days prior notice and subject to Section 1(d), delivery to the Company of
certificate(s) representing a number of shares of Preferred Stock having an
aggregate Preferred Value equal to the aggregate Exercise Price for the number
of Warrants being exercised. If the aggregate Preferred Value of the Preferred
Stock delivered in payment of the aggregate Exercise Price exceeds (because of
fractional shares) the aggregate Exercise Price for the number of Warrants being
exercised; then (subject to Section 1(d)) the Company will promptly pay to the

                                       2
<PAGE>

holder of the Warrants in cash such excess amount; provided that such excess
amount shall in no event be more than the Preferred Value of one share of
Preferred Stock. If the holder of this Warrant Certificate at any time exercises
less than all the Warrants, the Company shall issue to such a holder a warrant
certificate identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number of Warrants previously exercised.
Likewise, upon the presentation and surrender of this Warrant Certificate to the
Company at its address set forth in SECTION 10 and at the request of the holder,
the Company will, without expense, at the option of the holder, issue to the
holder in substitution for this Warrant Certificate one or more warrant
certificates in identical form and for an aggregate number of Warrants equal to
the number of Warrants evidenced by this Warrant Certificate.

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

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

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

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

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

                                       3
<PAGE>

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

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

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

                                       4
<PAGE>

         (d) CERTAIN ISSUANCES OF SECURITIES. Subject to SECTION 2(f), if the
Company at any time after the Issuance Date shall issue any additional shares of
Common Stock (otherwise than as provided in subsections (a) through (c) of this
SECTION 2) at a price per share less than the Average Price per share of Common
Stock for the 20 trading days immediately preceding the date of the
authorization of such issuance (the "Market Price") by the Board of Directors or
its compensation committee (as applicable), then the Exercise Price upon each
such issuance shall be adjusted to that price determined by multiplying the
Exercise Price by a fraction:

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

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

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

         (e) COMMON STOCK EQUIVALENTS.

                  i. Subject to SECTION 2(f), if the Company shall, after the
         Issuance Date, issue any security or evidence of indebtedness which is
         convertible into or exchangeable for Common Stock ("Convertible
         Security"), or any warrant, option or other right to subscribe for or
         purchase Common Stock or any Convertible Security, other than pursuant
         to Employee Benefit Plans (together with Convertible Securities,
         "Common Stock Equivalent"), then the Exercise Price upon each such
         issuance shall be adjusted as provided in SECTION 2(d) on the basis
         that (i) the maximum number of additional shares of Common Stock

                                       5
<PAGE>

         issuable pursuant to all such Common Stock Equivalents shall be deemed
         to have been issued as of the date of issuance of such Common Stock
         Equivalent; and (ii) the aggregate consideration for such maximum
         number of additional shares of Common Stock shall be deemed to be the
         minimum consideration received and receivable by the Company for the
         issuance of such additional shares of Common Stock pursuant to such
         Common Stock Equivalent.

                  ii. Notwithstanding the foregoing, no adjustment shall be made
         pursuant to this SECTION 2(e) unless the consideration received and
         receivable by the Company per share of Common Stock for the issuance of
         such additional shares of Common Stock pursuant to such Common Stock
         Equivalent is less than the Market Price. No adjustment of the Exercise
         Price shall be made under this SECTION 2(e) upon the issuance of any
         Convertible Security which is issued pursuant to the exercise of any
         warrants or other subscription or purchase rights therefor, if any
         adjustment shall previously have been made in the Exercise Price then
         in effect upon the issuance of such warrants or other rights pursuant
         to this SECTION 2(e). No adjustment shall be made under this SECTION
         2(e) if an adjustment is to be made under SECTION 2(f). No adjustment
         shall be made as a result of adjustment in the exercise or conversion
         price of Common Stock Equivalents, if those adjustments occur by the
         terms of such Common Stock Equivalents.

         (f) SPECIAL ADJUSTMENTS OF EXERCISE PRICE. Notwithstanding anything to
the contrary in SECTION 2(d) or SECTION 2(e), this SECTION 2(f) shall govern
adjustments to the Exercise Price for the transactions described in this SECTION
2(f).

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

                  ii. If the Company at any time after the Issuance Date and
         prior to the second anniversary of the Issuance Date, issues any Common
         Stock Equivalent (which by definition excludes Employee Benefit Plan
         securities) (otherwise than as provided in subsections (a) through (c)
         of Section 2; or pursuant to any Common Stock Equivalent outstanding as
         of the Issuance Date) and the New CSE Exercise Price (defined below) of
         such Common Stock Equivalents is less than the Exercise Price then in
         effect, then the Exercise Price upon each such issuance shall be
         adjusted to the New CSE Exercise Price of such Common Stock
         Equivalents. The "New CSE Exercise Price" shall be determined by
         dividing (x) the total amount, if any, received or receivable by the
         Company as consideration for the issuance of such Common Stock
         Equivalents, plus the minimum aggregate amount of additional
         consideration payable to the Company upon the exercise, conversion or
         exchange of such Common Stock Equivalents, plus, in the case of any
         such Common Stock Equivalents which relate to Convertible Securities,
         the minimum aggregate amount of additional consideration, if any,
         payable to the Company upon the conversion or exchange of such
         Convertible Securities, by (y) the total maximum number of shares of
         Common Stock issuable upon the exercise, conversion or exchange of all
         such Common Stock Equivalents.

                                       6
<PAGE>

         (g) MISCELLANEOUS. The following provisions shall be applicable to the
making of adjustments in the Exercise Price hereinbefore provided in this
SECTION 2:

                  i. The consideration received by the Company shall be deemed
         to be the following: (I) to the extent that any additional shares of
         Common Stock or any Common Stock Equivalent shall be issued for cash
         consideration, the consideration received by the Company therefor, or,
         if such additional shares of Common Stock or Common Stock Equivalent
         are offered by the Company for subscription, the subscription price,
         or, if such additional shares of Common Stock or Common Stock
         Equivalent are sold to underwriters or dealers for public offering
         without a subscription offering, the initial public offering price, in
         any such case excluding any amounts paid or receivable for accrued
         interest or accrued dividends and without deduction of any
         compensation, discounts, commissions or expenses paid or incurred by
         the Company for and in the underwriting of , or otherwise in connection
         with, the issue thereof; (II) to the extent that such issuance shall be
         for a consideration other than cash, then, except as herein otherwise
         expressly provided, the fair value of such consideration at the time of
         such issuance as determined in good faith by the Board of Directors, as
         evidenced by a certified resolution of the Board of Directors delivered
         to the holder of this Warrant Certificate setting forth such
         determination. The consideration for any additional shares of Common
         Stock issuable pursuant to any Common Stock Equivalent shall be the
         consideration received by the Company for issuing such Common Stock
         Equivalent, plus the additional consideration payable to the Company
         upon the exercise, conversion or exchange of such Common Stock
         Equivalent. In case of the issuance at any time of any additional
         shares of Common Stock or Common Stock Equivalent in payment or
         satisfaction of any dividend upon any class of stock other than Common
         Stock, the Company shall be deemed to have received for such additional
         shares of Common Stock or Common Stock Equivalent (which shall not be
         deemed to be a dividend payable in, or other distribution of, Common
         Stock under SECTION 2(c) above) consideration equal to the amount of
         such dividend so paid or satisfied. In the event additional shares of
         Common Stock or Common Stock Equivalents are issued together with other
         shares or securities or other assets of the Company or its subsidiaries
         for consideration which covers both, the consideration for such shares
         of Common Stock and Common Stock Equivalents shall be computed based on
         the respective portions of such consideration so received, computed as
         provided in this SECTION 2(g)i., as determined and allocated in good
         faith by the Board of Directors of the Company.

                  ii. Upon the expiration of the right to convert, exchange or
         exercise any Common Stock Equivalent the issuance of which effected an
         adjustment in the Exercise Price, if any such Common Stock Equivalent
         shall not have been converted, exercised or exchanged, the number of
         shares of Common Stock deemed to be issued and outstanding because they

                                       7
<PAGE>

         were issuable upon conversion, exchange or exercise of any such Common
         Stock Equivalent shall no longer be computed as set forth above, and
         the Exercise Price shall forthwith be readjusted and thereafter be the
         price which it would have been (but reflecting any other adjustments in
         the Exercise Price made pursuant to the provisions of SECTION 2(d)
         after the issuance of such Common Stock Equivalent) had the adjustment
         of the Exercise Price made upon the issuance or sale of such Common
         Stock Equivalent been made on the basis of the issuance only of the
         number of additional shares of Common Stock actually issued upon
         exercise, conversion or exchange of such Common Stock Equivalent and
         thereupon only the number of additional shares of Common Stock actually
         so issued shall be deemed to have been issued and only the
         consideration actually received by the Company (computed as in this
         SECTION 2(f)(i) shall be deemed to have been received by the Company.

                      III. THE NUMBER OF SHARES OF COMMON STOCK AT ANY TIME
         OUTSTANDING SHALL NOT INCLUDE ANY SHARES THEREOF THEN DIRECTLY OR
         INDIRECTLY OWNED OR HELD BY OR FOR THE ACCOUNT OF THE COMPANY OR ITS
         WHOLLY OWNED SUBSIDIARIES.

                      IV. UPON EACH ADJUSTMENT OF THE EXERCISE PRICE AS A RESULT
         OF THE CALCULATIONS MADE IN SECTION 2(d), (e) AND (f) HEREOF, THIS
         WARRANT SHALL THEREAFTER EVIDENCE THE RIGHT TO PURCHASE, AT THE
         ADJUSTED EXERCISE PRICE, THAT NUMBER OF SHARES OF COMMON STOCK OBTAINED
         BY (i) MULTIPLYING THE NUMBER OF SHARES COVERED BY THIS WARRANT
         IMMEDIATELY PRIOR TO SUCH ADJUSTMENT OF THE NUMBER OF SHARES BY THE
         EXERCISE PRICE IN EFFECT IMMEDIATELY PRIOR TO SUCH ADJUSTMENT OF THE
         EXERCISE PRICE AND (ii) DIVIDING THE PRODUCT SO OBTAINED BY THE
         EXERCISE PRICE IN EFFECT IMMEDIATELY AFTER SUCH ADJUSTMENT OF THE
         EXERCISE PRICE.

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

         (h) CALCULATION OF EXERCISE PRICE. The Exercise Price in effect from
time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

                                       8
<PAGE>

         (i) NASDAQ MATTERS. Notwithstanding anything to the contrary herein,
any adjustment to the Exercise Price that would require shareholder approval
pursuant to the Nasdaq Market Rules shall be subject to the Company's obtaining
such requisite approval.

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

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

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

         Section 5.        MANDATORY EXERCISE.

         (a) If (i) the Price of the Common Stock is greater than $5.00 per
share (as adjusted to reflect any stock split, combination, reclassification,
recapitalization, exchange, stock dividend or other distribution payable in
Common Stock with respect to shares of Common Stock) for sixty (60) consecutive
trading days in the principal market in which the Common Stock is traded and
(ii) the Company gives written notice (the "Company Notice") to the holder
hereof of the satisfaction of the condition in clause (i), then within fifteen
(15) days after the effective date of the Company Notice, the holder hereof
shall exercise all of the Warrants. If required by this SECTION 5, the holder
hereof agrees to exercise the Warrants, and to purchase shares of Common Stock
pursuant to the terms of this Warrant Certificate. If the holder has not
fulfilled its obligations to exercise the Warrants pursuant to this Section 5
within fifteen (15) days after the holder's receipt of the Company Notice, then
(without limiting the Company's available remedies) (A) the obligations of
holder under this Section 5 shall continue but the purchase rights otherwise
represented by this Warrant Certificate shall terminate, (B) the Company may
thereafter refuse, in its sole discretion, to allow holder to exercise the
Warrants (including pursuant to this Section 5), (C) all obligations of the
Company under Sections 3, 6, 7 and 8 shall terminate, (D) no further adjustments
to the Exercise Price shall be made unless the Company in its sole discretion
consents in writing. Each Warrant holder's obligations under this Section 5(a)
shall be subject to the expiration or termination of all waiting periods (and
any extensions thereof) applicable to exercise of such holder's Warrants under
the HSR Act (as defined below); provided that such holder shall have certified
in writing to the Company that a filing under the HSR Act is required and
provided further that such holder shall use its best efforts to cause the
expiration or termination of such waiting period to occur as promptly as
practicable.

                                       9
<PAGE>

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

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

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

         Section 6.        NOTICES TO WARRANT HOLDERS.  In the event:

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

         (b) the Company shall declare any dividend (or any other distribution)
on the Common Stock or any other class of its capital stock, other than
dividends on the Shares, as defined in the Securities Purchase Agreement; or

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

                                       10
<PAGE>

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

         Section 7. REPORTS TO WARRANT HOLDERS. The Company will cause to be
delivered, by first-class mail, postage prepaid, to holder at such holder's
address appearing hereon, or such other address as the holder shall specify, a
copy of any reports delivered by the Company to the holders of Common Stock.

         Section 8. COVENANTS OF THE COMPANY. The Company covenants and agrees
that:

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

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

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

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

                                       11
<PAGE>

                  (e) As soon as practicable after the receipt from the holder
of this Warrant Certificate of notice of the intent to exercise of a number of
warrants sufficient to require a filing under the Hart-Scott-Rodino Antitrust
Improvements Act of 1976 and the rules, regulations and formal interpretations
thereunder, as amended from time to time (the "HSR Act") (and after the receipt,
if applicable, of the notice referred to in Rule 803.5 of the HSR Act), but in
any event no later than the 15th business day after receipt of such notice(s),
the Company will (i) if required by the HSR Act, prepare and file with Antitrust
Division of the Department of Justice (the "DOJ") and the Federal Trade
Commission (the "FTC") the Notification and Report Form (accompanied by all
documentary attachments contemplated thereby) required by the HSR Act, (ii) upon
request of the holder, request early termination of the waiting period imposed
by the HSR Act, and (iii) coordinate and cooperate with the holder in responding
to formal and informal requests for additional information and documentary
material from the DOJ and the FTC in connection with such filing.
Notwithstanding the foregoing, if the holder is required to file with the DOJ
and FTC the Notification and Report Form solely as a result of its holding
and/or purchasing shares of Common Stock issued pursuant to this Warrant (with
no regard to any other securities held by such holder or its affiliates) and the
holder certifies such fact to the Company in writing, the Company agrees to
promptly reimburse the holder for all fees and expenses for the preparation and
filing of such form, including all legal expenses and filing fees.

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

         Section 9. NO RIGHTS AS STOCKHOLDER. The holder of the Warrants shall
not, by virtue of holding such Warrants, be entitled to any rights of a
stockholder of the Company either at law or in equity, and the rights of the
holder of the Warrants are limited to those expressed herein.

         Section 10. NOTICES. All notices, requests, demands, and other
communications required or permitted to be given or made hereunder by any party
hereto shall be in writing and shall be deemed to have been duly given or made
if (i) delivered personally, (ii) sent by prepaid overnight courier service, or
(iii) sent by telecopy or facsimile transmission, answer back requested, to the
parties at the following addresses (or at such other addresses as shall be
specified by the parties by like notice):

                  if to the holder, to such holder at:

                           277 Park Avenue
                           New York, New York 10172
                           Attention: Michael Isikow
                           Telefax: 212-892-2689

                  with a copy to:

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

                  and, if to the Company:

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

                                       12
<PAGE>

Such notices, requests, demands, and other communications shall be effective (i)
if delivered personally or sent by courier service, upon actual receipt by the
intended recipient, or (ii) if sent by telecopy or facsimile transmission, when
the answer back is received.

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

         Section 12. LOST, STOLEN, MUTILATED OR DESTROYED WARRANT CERTIFICATES.
Upon receipt by the Company of evidence reasonably satisfactory to it of the
ownership of and the loss, theft, destruction or mutilation of any Warrant
Certificate, then, in the absence of notice to the Company that such Warrant
Certificate has been acquired by a bona fide purchaser, the Company shall
execute and deliver, in exchange for or in lieu of the lost, stolen, destroyed
or mutilated Warrant Certificate, a substitute Warrant Certificate of the same
tenor and evidencing a like number of Warrants.

         Section 13. TRANSFER. Subject to Section 14 and the Securities Purchase
Agreement, transfer of Warrants, in whole or in part, shall be registered on the
books of the Company to be maintained for such purposes, upon surrender of the
Warrant Certificate representing such Warrants at the principal office of the
Company referred to in SECTION 10, together with a written assignment
substantially in the form of Exhibit B to this Warrant Certificate and a written
agreement, in form reasonably satisfactory to the Company, setting forth the new
Warrant holder's agreement to be bound by all of the terms of this Warrant
Certificate (including without limitation Section 14) and Section 5.5 of the
Securities Purchase Agreement, each duly executed by the holder, and funds
sufficient to pay any transfer taxes payable by such holder upon the making of
such transfer. Upon such surrender and, if required, such payment, the Company
shall execute and deliver a new Warrant Certificate or Warrant Certificates in
the name of the assignee or assignees and in the denomination specified in such
instrument of assignment, and shall issue to the assignor a new Warrant
Certificate or Warrant Certificates evidencing the portion of the old Warrant
Certificate not so assigned, and the old Warrant Certificate shall promptly be
canceled.

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

                         *            *            *

              IN WITNESS WHEREOF, the Company has caused this Warrant
Certificate to be executed as of November 1, 2000, by the undersigned, thereunto
duly authorized.

                                BRIGHAM EXPLORATION COMPANY

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

                                DLJ MB FUNDING III, INC.

                                By: /s/ IVY B. DODES
                                    -------------------------------------
                                        Ivy B. Dodes
                                        Principal

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