<PAGE>   1
                                                                     EXHIBIT 4.5


                          REGISTRATION RIGHTS AGREEMENT

         This Registration Rights Agreement (this "Agreement") is made and
entered into as of ______________, 1998, by and among Brigham Exploration
Company, a Delaware corporation (the "Company"), _______________________________
________________________________________, a ___________________ ("_____"), and
____________________________________, a ____________________ ("____").

         This Agreement is made pursuant to the Securities Purchase Agreement
(the "Securities Purchase Agreement") dated as of ______________, 1998, by and
among the Company, ____ and ____. In order to induce ____ and ____ to enter into
the Securities Purchase Agreement, the Company has agreed to provide the
registration and other rights set forth in this Agreement. Pursuant to the
Securities Purchase Agreement, ____ and ____ will purchase the Acquired Shares
and Warrants. The execution and delivery of this Agreement is a condition to the
Closing (as defined in the Securities Purchase Agreement).

         The parties agree as follows:

                                    ARTICLE I

         Section 1.01. Definitions. Capitalized terms used herein without
definition shall have the meanings given to them in the Securities Purchase
Agreement. The terms set forth below are used herein as so defined:

              "Commission" has the meaning specified therefor in Section 1.02 of
this Agreement.

              "Common Stock" means the common stock, par value $0.01 per share,
of the Company.

              "Exchange Act" shall mean the Securities Exchange Act of 1934, as
amended, and the rules and regulations promulgated thereunder.

              "Holder" means the record holder of any Registrable Securities.

              "Inspector" has the meaning specified therefor in Section 2.03
this Agreement.

              "Losses" has the meaning specified therefor in Section 2.07 of
this Agreement.

              "Other Holders" has the meaning specified therefor in Section 2.01
of this Agreement.

              "Person" means any individual, corporation, limited liability
company, partnership, joint venture, association, joint-stock company, business
trust, trust or unincorporated entity.

              "Records" has the meaning specified therefor in Section 2.03 of
this Agreement.




<PAGE>   2

              "Registrable Securities" means the Acquired Shares, the Warrants,
the Warrant Shares and any other securities issued upon the exercise of the
Warrants, until such time as such securities cease to be Registrable Securities
pursuant to Section 1.02 hereof.

              "Requesting Holder(s)" has the meaning specified therefor in
Section 2.01 of this Agreement.

              "Request Notice" has the meaning specified therefor in Section
2.01 of this Agreement.

              "Registration Statement" has the meaning specified therefor in
Section 2.01 of this Agreement.

              "Securities Act" has the meaning specified therefor in Section
1.02 of this Agreement.

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

              "Selling Holder" means a Holder who is selling Registrable
Securities pursuant to a Registration Statement.

         Section 1.02. Registrable Securities. Any Registrable Security will
cease to be a Registrable Security when (i) a Registration Statement covering
such Registrable Security has been declared effective by the Securities and
Exchange Commission (the "Commission") and such Registrable Security has been
sold or disposed of pursuant to such effective Registration Statement; (ii) such
Registrable Security is disposed of pursuant to Rule 144 (or any similar
provision then in force) under the Securities Act of 1933, as amended (the
"Securities Act"); (iii) such Registrable Security is eligible to be, and at the
time of determination can be, disposed of pursuant to paragraph (k) of Rule 144
(or any similar provision then in force) under the Securities Act; or (iv) such
Registrable Security is held by the Company or one of its subsidiaries.

                                   ARTICLE II

         Section 2.01. Demand Registration. (a) Any time after the Funding Date,
as defined in the Securities Purchase Agreement, (i) any Holder or Holders who
collectively beneficially own at least 25% (twenty-five percent) of the
Registrable Securities may request (a "Request Notice") the Company to register
under the Securities Act (other than pursuant to a shelf registration on Form
S-3) all or any portion of the Registrable Securities that are held by such
Holder or Holders (collectively, the "Requesting Holder") for sale in the manner
specified in the Request Notice and (ii) in the event that the Company is
eligible to file a Registration Statement on Form S-3, any Holders who
collectively beneficially own at least a majority of the Registrable Securities
may submit a Request Notice to the Company to register under Form S-3 all or any
portion of the Registrable Securities that are held by such Requesting Holder(s)
for sale in the manner specified in the Request Notice.



                                       -2-

<PAGE>   3

                  (b) Promptly following receipt of a Request Notice, the
Company shall immediately notify each Holder (except the Requesting Holder) of
the receipt of a Request Notice and shall use its best efforts to file a
registration statement under the Securities Act (each such registration
statement is hereinafter referred to as a "Registration Statement") effecting
the registration under the Securities Act, for public sale in accordance with
the method of disposition specified in such Request Notice, of the Registrable
Securities specified in the Request Notice (and in any notices that the Company
receives from other Holders no later than the 15th day after receipt of the
notice sent by the Company) (such other Holders and the Requesting Holder are
hereinafter referred to as the "Requesting Holders"). If such method of
disposition shall be an underwritten public offering, the Requesting Holders
holding a majority of the Registerable Securities to be registered may designate
the managing underwriter of such offering, subject to the approval of the
Company, which approval shall not be withheld unreasonably. The Company shall be
obligated to register Registrable Securities pursuant to this Section 2.01 on
two occasions only. A request pursuant to this Section 2.01 shall be counted
("Counted") only when (i) all the Registrable Securities requested to be
included in any such registration have been so included, (ii) the corresponding
Registration Statement has become effective under the Securities Act, and (iii)
the public offering has been consummated and the Registrable Securities have
been sold on the terms and conditions specified therein; provided, however, that
in the event of a shelf registration if the Company is then eligible to file a
Registration Statement on Form S-3, the Company shall keep such Registration
Statement effective for two years from the effective date of the Registration
Statement.

                  (c) The Company may delay the filing or effectiveness of a
Registration Statement after receipt of a Request Notice (i) for up to 90 days
if at the time of such request, the Company is engaged in a firm commitment
underwritten public offering of its securities in which Holders may include
Registrable Securities and for which the Company has delivered the notice to the
Holders required by Section 2.02 or (ii) for up to 60 days if at the time of
such request the Board of Directors of the Company determines in its reasonable
judgment and in good faith that the filing of such a Registration Statement or
the making of any required disclosure in connection therewith would have a
material adverse effect on the Company or substantially interfere with a
significant transaction in which the Company is then engaged; provided that the
Company may not delay the filing of a Registration Statement in reliance on this
clause (ii) more than once during any period of twelve consecutive calendar
months.

                  (d) The Company and any institutional investor who is entitled
to piggy-back registration rights as of the Closing Date ("Other Holders") with
respect to a Registration Statement filed pursuant to Section 2.01 may include
securities of the Company in such Registration Statement, but only to the extent
that, in the good faith opinion of the managing underwriter (if such method of
disposition shall be an underwritten public offering) supported by written
reasons therefor, the inclusion of such shares would not raise a substantial
doubt as to whether the proposed offering could successfully be consummated.
Subject to Section 2.01(c) above, except as provided in this subsection (d), the
Company will not effect any other registration of its equity securities (except
with respect to Registration Statements on Form S-4 or S-8 or for purposes
permissible under such forms as of the date hereof), whether for its own account
or that of any Other Holder, from the date of receipt of a Request Notice
related to an underwritten public offering until the completion of the
distribution by the underwriters of all securities thereunder.



                                       -3-

<PAGE>   4

                  (e) Exercise of the right to convert the Warrants to Warrant
Shares shall at the Holder's sole election and discretion be contingent upon the
registration of the Warrant Shares in accordance with this Agreement and should
such registration not be completed pursuant to the terms hereof, Holder shall
have the right, at its sole discretion, to rescind its election to convert the
Warrants.

                  (f) From and after the date of this Agreement and until no
Registrable Securities remain outstanding, the Company shall not grant
registration rights to any Person if such rights would conflict with the
provisions of this Agreement.

                  Section 2.02. Piggy-Back Registration. If the Company proposes
to register any of its equity securities under the Securities Act for sale to
the public, whether for its own account or for the account of Other Holders or
both (except with respect to Registration Statements on Forms S-4 or S-8 or for
purposes permissible under such forms as of the date hereof), each such time it
will give written notice to all Holders of its intention to do so no less than
20 days prior to the anticipated filing date. Upon the written request received
by the Company from any Holder no later than the 15th day after receipt by such
Holder of the notice sent by the Company (which request shall state the intended
method of disposition thereof), the Company will use best efforts to cause the
Registrable Securities as to which registration shall have been so requested to
be included in the securities to be covered by such Registration Statement, all
to the extent requisite to permit the sale or other disposition by each Holder
(in accordance with its written request) of such Registrable Securities so
registered; provided, however, that the Company may at any time prior to the
effectiveness of any such Registration Statement, in its sole discretion and
without the consent of any Holder, abandon any proposed offering by the Company
in which any Holder had requested to participate. The number of Registrable
Securities to be included in such a registration may be reduced or eliminated if
and to the extent, in the case of an underwritten offering, the managing
underwriter shall advise the Company that such inclusion would materially
jeopardize the successful marketing of the securities (including the Registrable
Securities) proposed to be sold therein; provided, however, that such number of
shares of Registrable Securities shall not be reduced if any securities included
in such registration are included other than for the account of the Company
unless the shares included in the Registration for the account of such Persons
are also reduced on a pro rata basis, provided, in the case of a Registration
Statement filed pursuant to the exercise of demand registration rights of any
Other Holders, priority shall be given first to the Other Holders demanding such
registration.

                  Section 2.03. Registration Procedures. If and whenever the 
Company is required pursuant to this Agreement to effect the registration of 
any of the Registrable Securities under the Securities Act, the Company will, as
expeditiously as possible:

                  (a) prepare and file with the Commission a Registration
Statement, on a form available to the Company, with respect to such securities
(which filing shall be made (i) as promptly as possible (but in no event later
than 30 days after the receipt by the Company of a Request Notice) in the case
of a shelf registration if the Company is then eligible to file a Registration
Statement on Form S-3 or (ii) as promptly as possible (but in no event later
than 60 days after the receipt by the Company of a Request Notice) in the case
of any underwritten offering or if the Company is not


                                       -4-

<PAGE>   5

eligible to file a Registration Statement on Form S-3, provided that in no event
will the Company be obligated to file a Registration Statement pursuant to (i)
or (ii) prior to the Funding Date pursuant to the Securities Purchase Agreement.
The Company shall thereafter use best efforts to cause such Registration
Statement to become and remain effective for the period of the distribution in
order for it to be Counted;

                  (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 for
the distribution period to be Counted and as may be necessary to comply with the
provisions of the Securities Act with respect to the disposition of all
securities covered by such Registration Statement in accordance with the Holders
intended method of disposition;

                  (c) furnish to each Selling Holder and to each underwriter
such number of copies of the Registration Statement and the prospectus included
therein (including each preliminary prospectus and each document incorporated by
reference therein to the extent then required by the rules and regulations of
the Commission) as such Persons may reasonably request in order to facilitate
the public sale or other disposition of the Registrable Securities covered by
such Registration Statement;

                  (d) if applicable, use best efforts to register or qualify the
Registrable Securities covered by such Registration Statement under the
securities or blue sky laws of such jurisdictions as the Selling Holders or, in
the case of an underwritten public offering, the managing underwriter, shall
reasonably request, provided that the Company will not be required to qualify
generally to transact business in any jurisdiction where it is not then required
to so qualify or to take any action that would subject it to general service of
process in any such jurisdiction where it is not then so subject;

                  (e) immediately notify each Selling Holder and each
underwriter, at any time when a prospectus relating thereto is required to be
delivered under the Securities Act, of the happening of any event as a result of
which the prospectus contained in such Registration Statement, as then in
effect, includes an untrue statement of a material fact or omits to state any
material fact required to be stated therein or necessary to make the statements
therein not misleading in the light of the circumstances then existing and as
promptly as practicable amend or supplement the prospectus or take other
appropriate action so that the prospectus does not include an untrue statement
of a material fact or omit 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;

                  (f) in the case of an underwritten public offering, furnish
upon request, (i) on the date that Registrable Securities are delivered to the
underwriters for sale pursuant to such Registration Statement, an opinion of
counsel for the Company dated as of such date and addressed to the underwriters
and to the Selling Holders, stating that such Registration Statement has become
effective under the Securities Act and that (A) to the best knowledge of such
counsel, no stop order suspending the effectiveness thereof has been issued and
no proceedings for that purpose have been instituted or are pending or
contemplated under the Securities Act, (B) the Registration Statement,



                                       -5-

<PAGE>   6

the related prospectus, and each amendment or supplement thereof, comply as to
form in all material respects with the requirements of the Securities Act and
the applicable rules and regulations thereunder of the Commission (except that
such counsel need express no opinion as to the financial statements, or any
expertized schedule, report or information contained or incorporated therein)
and (C) to such other effects as may reasonably be requested by counsel for the
underwriters, and (ii) on the effective date of the Registration Statement and
on the date that Registrable Securities are delivered to the underwriters for
sale pursuant to such Registration Statement, a letter dated such dates from the
independent accountants retained by the Company, addressed to the underwriters
and, if available, to the Selling Holders, stating that they are independent
public accountants within the meaning of the Securities Act and that, in the
opinion of such accountants, the financial statements of the Company and the
schedules thereto that are included or incorporated by reference in the
Registration Statement or the prospectus, or any amendment or supplement
thereof, comply as to form in all material respects with the applicable
requirements of the Securities Act and the published rules and regulations
thereunder, and such letter shall additionally address such other financial
matters (including information as to the period ending no more than five
business days prior to the date of such letter) included in the Registration
Statement in respect of which such letter is being given as the underwriters may
reasonably request;

                  (g) make available for inspection by the Selling Holders
designated by a majority thereof, any underwriter participating in any
distribution pursuant to such Registration Statement, and any attorney,
accountant or other agent retained by such representative of the Selling Holders
or underwriter (the "Inspectors"), all financial and other records, pertinent
corporate documents and properties of the Company (collectively, the "Records"),
and cause the Company's officers, directors and employees to supply all
information reasonably requested by any such Inspector in connection with such
Registration Statement.

                  (h) cause the Registerable Securities to be listed on New York
Stock Exchange, American Stock Exchange or on the NASDAQ National Market if the
Common Stock is or becomes so listed;

                  (i) use best efforts to keep effective and maintain for the
period of distribution to be Counted, qualification, approval or listing
obtained to cover the Registrable Securities as may be necessary for the Selling
Holders to dispose thereof and shall from time to time amend or supplement any
prospectus used in connection therewith to the extent necessary in order to
comply with applicable law;

                  (j) use best efforts to cause the Registrable Securities to be
registered with or approved by such other governmental agencies or authorities
as may be necessary by virtue of the business and operations of the Company to
enable the Selling Holders to consummate the disposition of such Registrable
Securities; and

                  (k) take such other actions as are reasonably requested by the
Selling Holders or the underwriters, if any, in order to expedite, facilitate or
consummate the disposition of such Registrable Securities.



                                       -6-

<PAGE>   7

         In connection with each registration hereunder with respect to an
underwritten public offering, the Company and each Selling Holder agrees to
enter into a written agreement with the managing underwriter or underwriters
selected in the manner herein provided in such form and containing such
provisions as are customary in the securities business for such an arrangement
between underwriters and companies of the Company's size and investment stature,
provided that such agreement shall not contain any such provision applicable to
the Company or the Selling Holders that is inconsistent with the provisions
hereof; and further provided, that the time and place of the closing under said
agreement shall be as mutually agreed upon among the Company, the Selling
Holders and such managing underwriter. In connection with each registration
hereunder, each Selling Holder will furnish promptly to the Company in writing
such information with respect to itself and the proposed distribution by it as
shall be necessary in order to ensure compliance with federal and applicable
state securities laws.

         Section 2.04. Restrictions on Public Sale by Selling Holders of
Registrable Securities. To the extent not inconsistent with applicable law,
including insurance codes, each Selling Holder whose Registrable Securities are
included in a Registration Statement pursuant to this Agreement agrees not to
effect any public sale or distribution of the issue being registered (or any
securities of the Company convertible into or exchangeable or exercisable for
securities of the same type as the issue being registered) during the 14 days
before, and during the 90-day period beginning on, the effective date of such
Registration Statement (except as part of such registration), but only if and to
the extent requested in writing (with reasonable prior notice) by the managing
underwriter or underwriters in the case of an underwritten public offering by
the Company of securities of the same type as the Registrable Securities,
provided that the duration of the foregoing restrictions shall be no longer than
the duration of the shortest restriction imposed by the underwriters on the
officers or directors or any other stockholder of the Company; and provided
further, that the period of time for which the Company is required to keep such
registration statement which includes Registrable Securities continuously
effective shall be increased by a period equal to such requested holdback
period.

         Section 2.05. Restrictions on Public Sale by the Company. To the extent
required by an underwriter in an underwritten public offering, the Company
agrees not to effect on its own behalf any public sale or distribution of any
securities similar to those being registered, or any securities convertible into
or exchangeable or exercisable for such securities, during the 14 days before,
and during the 90-day period beginning on, the effective date of any
registration statement in which the Selling Holders of Registrable Securities
are participating, other than pursuant to such registration statement or a
Registration Statement on Form S-8 or Form S-4.

         Section 2.06.     Expenses.

                  (a) "Registration Expenses" means all expenses incident to the
Company's performance under or compliance with this Agreement, including without
limitation, all registration and filing fees, blue sky fees and expenses,
printing expenses, listing fees, fees and disbursements of counsel and
independent public accountants for the Company, fees of the National Association
of Securities Dealers, Inc., transfer taxes, fees of transfer agents and
registrars, costs of insurance and reasonable out-of-pocket expenses (including
without limitation, reasonable legal fees of one counsel



                                       -7-

<PAGE>   8

for all Selling Holders), but excluding any Selling Expenses. "Selling Expenses"
means all underwriting fees, discounts and selling commissions allocable to the
sale of the Registrable Securities.

                  (b) The Company will pay all Registration Expenses in
connection with each Registration Statement filed pursuant to this Agreement,
whether or not the Registration Statement becomes effective, and the Selling
Holders shall pay all Selling Expenses in connection with any Registrable
Securities registered pursuant to this Agreement.

         Section 2.07. Indemnification. (a) In the event of a registration of
any Registrable Securities under the Securities Act pursuant to this Agreement,
the Company will indemnify and hold harmless each Selling Holder thereunder and
each underwriter, pursuant to the applicable underwriting agreement with such
underwriter, of Registrable Securities thereunder and each Person, if any, who
controls such Selling Holder or underwriter within the meaning of the Securities
Act and the Exchange Act, against any losses, claims, damages or liabilities
(including reasonable attorneys' fees) ("Losses"), joint or several, to which
such Selling Holder or underwriter or controlling Person may become subject
under the Securities Act, the Exchange Act or otherwise, insofar as such Losses,
(or actions in respect thereof) arise out of or are based upon any untrue
statement or alleged untrue statement of any material fact contained in any
Registration Statement under which such Registrable Securities were registered
under the Securities Act pursuant to this Agreement, any preliminary prospectus
or final prospectus contained therein, or any amendment or supplement thereof,
or arise out of or are based upon the omission or alleged omission to state
therein a material fact required to be stated therein or necessary to make the
statements therein not misleading, and will reimburse each such Selling Holder,
each such underwriter and each such controlling Person for any legal or other
expenses reasonably incurred by them in connection with investigating or
defending any such Loss or actions; provided, however, that the Company will not
be liable in any such case if and to the extent that any such loss, claim,
damage or liability arises out of or is based upon an untrue statement or
alleged untrue statement or omission or alleged omission so made in conformity
with information furnished by such Selling Holder, such underwriter or such
controlling Person in writing specifically for use in such Registration
Statement or prospectus.

                  (b) Each Selling Holder agrees to indemnify and hold harmless
the Company, its directors, officers, employees and agents and each Person, if
any, who controls the Company within the meaning of the Securities Act or of the
Exchange Act to the same extent as the foregoing indemnity from the Company to
such Selling Holder, but only with respect to information regarding such Selling
Holder furnished in writing by or on behalf of such Selling Holder expressly for
inclusion in any Registration Statement or prospectus relating to the
Registrable Securities, or any amendment or supplement thereto; provided,
however, that the liability of such Selling Holder shall not be greater in
amount than the dollar amount of the proceeds (net of any Selling Expenses)
received by such Selling Holder from the sale of the Registrable Securities
giving rise to such indemnification.

                  (c) Promptly after receipt by an indemnified party hereunder
of notice of the commencement of any action, such indemnified party shall, if a
claim in respect thereof is to be made against the indemnifying party hereunder,
notify the indemnifying party in writing thereof, but



                                       -8-

<PAGE>   9

the omission so to notify the indemnifying party shall not relieve it from any
liability which it may have to any indemnified party. In case any such action
shall be brought against any indemnified party and it shall notify the
indemnifying party of the commencement thereof, the indemnifying party shall be
entitled to participate in and, to the extent it shall wish, to assume and
undertake the defense thereof with counsel reasonably satisfactory to such
indemnified party and, after notice from the indemnifying party to such
indemnified party of its election so to assume and undertake the defense
thereof, the indemnifying party shall not be liable to such indemnified party
under this Section 2.07 for any legal expenses subsequently incurred by such
indemnified party in connection with the defense thereof other than reasonable
costs of investigation and of liaison with counsel as so elected; provided,
however, that, (i) if the indemnifying party has failed to assume the defense
and employ counsel or (ii) if the defendants in any such action include both the
indemnified party and the indemnifying party and counsel to the indemnified
party shall have concluded that there may be reasonable defenses available to
the indemnified party that are different from or additional to those available
to the indemnifying party or that the interests of the indemnified party
reasonably may be deemed to conflict with the interests of the indemnifying
party, then the indemnified party shall have the right to select a separate
counsel and to assume such legal defense and otherwise to participate in the
defense of such action, with the expenses and fees of such separate counsel and
other expenses related to such participation to be reimbursed by the
indemnifying party as incurred.

                  (d) If the indemnification provided for in this Section 2.07
is unavailable to the Company or the Selling Holders or is insufficient to hold
them harmless in respect of any Losses, then each such 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 as between the
Company on the one hand and each Selling Holder on the other, in such proportion
as is appropriate to reflect the relative fault of the Company on the one hand
and of each Selling Holder on the other in connection with the statements or
omissions which resulted in such Losses, as well as any other relevant equitable
considerations. The relative fault of the Company on the one hand and each
Selling Holder on the other shall be determined by reference to, among other
things, whether the untrue or alleged untrue statements of a material fact or
the omission or alleged omission to state a material fact has been made by, or
relates to, information supplied by such party, and the parties' relative
intent, knowledge, access to information and opportunity to correct or prevent
such statement or omission.

                  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 is not guilty of such fraudulent
misrepresentation.

                                   ARTICLE III


         Section 3.01. Communications. All notices and other communications
provided for or permitted hereunder shall be made in writing by telecopy,
courier service or personal delivery:

                                       -9-

<PAGE>   10

                  (a) if to a Holder, at the most current address given by such
Holder to the Company in accordance with the provisions of this Section 3.01,
which address initially is, with respect to JEDI or ECT, the address set forth
in the Securities Purchase Agreement, and

                  (b) if to the Company, initially at its address set forth in 
the Securities Purchase Agreement, and

                  (c) for each, thereafter at such other address, notice of
which is given in accordance with the provisions of this Section 3.01.

                  All such notices and communications shall be deemed to have
been received at the time delivered by hand, if personally delivered; when
receipt acknowledged, if telecopied; and on the next business day if timely
delivered to an air courier guaranteeing overnight delivery.

         Section 3.02. Successor and Assigns. This Agreement shall inure to the
benefit of and be binding upon the successors and assigns of each of the
parties, including subsequent holders of Registrable Securities.

         Section 3.03. Counterparts. This Agreement may be executed in any
number of counterparts and by different parties hereto in separate counterparts,
each of which counterparts, when so executed and delivered, shall be deemed to
be an original and all of which counterparts, taken together, shall constitute
but one and the same Agreement.

         Section 3.04. Headings. The headings in this Agreement are for
convenience of reference only and shall not limit or otherwise affect the
meaning hereof.

         Section 3.05. Governing Law. THE LAWS OF THE STATE OF TEXAS SHALL
GOVERN THIS AGREEMENT WITHOUT REGARD TO PRINCIPLES OF CONFLICT OF LAWS.

         Section 3.06. Severability of Provisions. Any provision of this
Agreement which is prohibited or unenforceable in any jurisdiction shall, as to
such jurisdiction, be ineffective to the extent of such prohibition or
unenforceability without invalidating the remaining provisions hereof or
affecting or impairing the validity or enforceability of such provision in any
other jurisdiction.

         Section 3.07. Entire Agreement. This Agreement, together with the
Securities Purchase Agreement and the other Basic Documents (as defined in the
Securities Purchase Agreement) are intended by the parties as a final expression
of their agreement and intended to be a complete and exclusive statement of the
agreement and understanding of the parties hereto in respect of the subject
matter contained herein. There are no restrictions, promises, warranties or
undertakings, other than those set forth or referred to herein with respect to
the registration rights granted by the Company with respect to the securities
sold pursuant to the Securities Purchase Agreement. This Agreement, the
Securities Purchase Agreement and the other Basic Documents supersede all prior
agreements and understandings between the parties with respect to such subject
matter.



                                      -10-

<PAGE>   11

         Section 3.08. Attorneys' Fees. In any action or proceeding brought to
enforce any provision of this Agreement, the successful party shall be entitled
to recover reasonable attorneys' fees in addition to its costs and expenses and
any other available remedy.

         Section 3.09. Amendment. This Agreement may be amended only by means of
a written amendment signed by the Company and by the Holders of a majority of
the Registrable Securities.

         Section 3.10. Registrable Securities Held by the Company or Its
Affiliates. In determining whether the Holders of the required amount of
Registrable Securities have concurred in any direction, amendment, supplement,
waiver or consent, Registrable Securities owned by the Company or one of its
Affiliates shall be disregarded.

         Section 3.11. Assignment of Rights. The rights of any Holder under this
Agreement may be assigned to any Person who acquires any Registrable Securities.
Any assignment of registration rights pursuant to this Section 3.11 shall be
effective only upon receipt by the Company of written notice from such assigning
Holder stating the name and address of any assignee. The rights of an assignee
under this Section 3.11 shall be the same rights granted to the assigning Holder
under this Agreement. In connection with any such assignment, the term "Holder"
as used herein shall, where appropriate to assign the rights and obligations of
the assigning Holder hereunder to such assignee, be deemed to refer to the
assignee.



                                      -11-

<PAGE>   12

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


                                            BRIGHAM EXPLORATION COMPANY



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







                                      -12-