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                                                                   EXHIBIT 10.42

                          REGISTRATION RIGHTS AGREEMENT


         This Registration Rights Agreement (this "Agreement") is made and
entered into as of March 30, 1999, by and among Brigham Exploration Company, a
Delaware corporation (the "Company"), and Veritas DGC Land, Inc., a Delaware
corporation ("Veritas").

         This Agreement is made pursuant to the Exchange Agreement between the
Company and Veritas of even date herewith (the "Exchange Agreement"). In order
to induce Veritas to enter into the Exchange Agreement, the Company has agreed
to provide the registration and other rights set forth in this Agreement.
Pursuant to the Exchange Agreement, the Company will issue Veritas shares of
Common Stock (as defined below).

         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 Exchange 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 therefore 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.

         "Registrable Securities" means the Acquired Shares until such time as
such securities cease to be Registrable Securities pursuant to Section 1.02
hereof.




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         "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.

         "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. Registration.

                  (a) The Company shall register on a Form S-3 (or Form S-1 or
successor form to either) Registration Statement all of the Registrable
Securities that are held by the Holder(s) so that such Registration Statement
is declared effective within 180 days of the closing under the Exchange
Agreement. The Company shall use its best efforts to keep such Registration
Statement effective until the second anniversary of such closing. With respect
to shares issued pursuant to the Exchange Agreement at the end of, or after,
such 180-day period, the Company shall register such shares on a Form S-3 (or
Form S-1 or successor form to either) Registration Statement within 30 days of
such shares' issuance. Each such Registration Statement referred to herein is a
"Registration Statement."

                  (b) The Company and the parties to the Registration Rights
Agreements with the Company dated February 26, 1997 and August 20, 1998,
respectively, who are entitled to piggy-back registration rights ("Other
Holders") with respect to a Registration Statement filed pursuant to Section
2.01 may include securities of the Company in such Registration Statement.

         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 filed pursuant to demand under
the Registration Rights Agreement dated August 20, 1998 and 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



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which registration shall have been so requested to be included in the
securities to be covered by such Registration Statement, all to the extent
required 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 that demand 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 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 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;

                  (b) 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;

                  (c) 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 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;

                  (d) immediately notify each Selling Holder, 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



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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;

                  (e) make available for inspection by the Selling Holders
designated by a majority thereof, and any attorney, accountant or other agent
retained by such representative of the Selling Holders (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;

                  (f) cause the Registrable 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;

                  (g) use best efforts to keep effective and maintain for the
period of distribution, 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;

                  (h) 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

                  (i) 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.

         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 in the
case of an underwritten public offering by the Company of securities of the
same type as the Registrable Securities, and even then only if and to the
extent requested in writing (with reasonable prior notice) by the managing
underwriter or underwriters 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.




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         Section 2.05. [Reserved]

         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 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
Person, if any, who controls such Selling Holder 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 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
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



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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 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 is 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.





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                                   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;

                  (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,

                  (b) if to the Company, initially at its address set forth in
its Form 10-K filed with the Commission, 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
Exchange Agreement and the other documents provided for therein 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 Exchange Agreement.
This Agreement, the Exchange



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Agreement and the other documents provided for therein supersede all prior
agreements and understandings between the parties with respect to such subject
matter except as specified in the Exchange Agreement.

         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.

         Section 3.12 Liquidated Damages. The parties hereto agree that upon
the failure of the Company to register shares of Registrable Securities held by
Veritas in the amounts and at the times required hereunder, and to meet its
obligations under Section 2.03(a) hereunder, Veritas shall be entitled to
liquidated damages from the Company in an amount equal to $0.25 per share of
Registrable Securities then held by Veritas. The parties agree that the
liquidated damages set forth herein are reasonable and not a penalty because of
the difficulty of ascertaining the actual damages that would be sustained by
Veritas in the event of such a material breach of this Agreement by the
Company. In the event of such a breach Veritas either may elect the liquidated
damages provided hereunder or, in the alternative, pursue its remedies at law.




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

                                  BRIGHAM EXPLORATION COMPANY


                                  By:   /s/ David T. Brigham
                                        ----------------------------------------
                                        Name: David T. Brigham 
                                             -----------------------------------
                                        Title: Vice President
                                              ----------------------------------

                                  VERITAS DGC LAND, INC.


                                  By:   /s/ Deanna Goodwin
                                        ----------------------------------------
                                        Name: Deanna Goodwin
                                             -----------------------------------
                                        Title: Vice President Finance
                                              ----------------------------------





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