Exhibit 10.51

SECOND AMENDMENT TO  REGISTRATION RIGHTS AGREEMENT

This Second Amendment to Registration Rights Agreement (this "Amendment") is entered into as of December 21, 2002, by and among Brigham Exploration Company, a
Delaware corporation (the "Company"), and the Credit Suisse First Boston entities listed on Schedule A hereto (the "CSFB Entities").

RECITALS

WHEREAS, pursuant to that certain Registration Rights Agreement, dated as of November 1, 2000 (the "Agreement"), by and among the Company, DLJMB Funding III,
Inc., a Delaware corporation, and DLJ ESC II, L.P., a Delaware limited partnership (collectively, the "November 2000 Investors"), the Company provided certain registration rights to the November 2000
Investors regarding shares of the Company's common stock issuable upon exercise of warrants issued by the Company to the November 2000 Investors;

WHEREAS,
pursuant to that certain First Amendment to Registration Rights Agreement, dated March 5, 2001 (the "March 2002 Amendment"), by and among the Company, each of the November 2000
Investors, DLJ Merchant Banking Partners III, L.P., a Delaware limited partnership, and DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners III, C.V., a
Netherlands Antilles limited partnership (collectively, the "March 2001 Investors"), the Company provided certain registration rights under the Agreement to the March 2001 Investors regarding shares
of the Company's common stock issuable upon exercise of warrants issued by the Company to the March 2001 Investors;

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

WHEREAS,
each of the March 2001 Investors is a party to a certain Warrant Certificate (the "March 2001 Warrant Certificates"), dated as of March 5, 2001, by and between the
Company and such investor;

WHEREAS,
each of DLJ Merchant Banking Partners III, L.P., a Delaware limited partnership; DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners
III, C.V., a Netherlands Antilles limited partnership; DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners III-1, C.V. and as attorney-in-fact for DLJ
Merchant Banking III, L.P., as Associate General Partner of DLJ Offshore Partners III-1, C.V., a Netherlands Antilles limited partnership; DLJ Merchant Banking III, Inc., as Advisory General Partner
on behalf of DLJ Offshore Partners III-2, C.V. and as attorney-in-fact for DLJ Merchant Banking III, L.P., as Associate General Partner of DLJ Offshore Partners III-2, C.V., a Netherlands Antilles
limited partnership; DLJ MB Partners III GmbH & Co. KG, a German limited partnership; Millennium Partners II, L.P., a Delaware limited partnership; and MBP III Plan Investors, L.P., a Delaware limited
partnership (collectively, the "December 2002 Investors, and together with the November 2000 Investors and the March 2001 Investors, the "Investors") is a party to a certain Warrant Certificate (the
"December 2002 Warrant Certificates," and together with the November 2000 Warrant Certificates and the March 2001 Warrant Certificates, the "Warrant Certificates"), dated as of December 20, 2002, by
and between the Company and such investor;

WHEREAS,
the December 2002 Warrant Certificates were executed and delivered in connection with the consummation of transactions contemplated by that certain Securities Purchase
Agreement, dated December 20, 2002, by and among the Company and the December 2002 Investors;

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WHEREAS,
pursuant to the Warrant Certificates, each Holder (as defined in the Agreement) has been issued a warrant (the "Warrant") to purchase shares of the Company's common stock, par
value $0.01 per share (the "Company's Common Stock");

WHEREAS,
the December 2002 Investors and Shell Capital Inc. ("SCI") are parties to that certain Assignment Agreement, dated as of November 21, 2002 (the "Assignment Agreement"), whereby
the December 2002 Investors have purchased from SCI a note payable by Brigham Oil & Gas, L.P. in the principal amount of $10,000,000 (the "Note") that is convertible into 2,564,102 shares of the
Company's Common Stock;

WHEREAS,
the Company and the December 2002 Investors are parties to that certain Omnibus Agreement, dated as of December 20, 2002 (the "Omnibus Agreement"), whereby the December 2002
Investors have converted the Note into 2,564,102 shares of the Company's Common Stock; and

WHEREAS,
to induce the Investors to enter into the Warrant Certificates, the Securities Purchase Agreement, and the Omnibus Agreement, the Company has agreed to provide registration
rights with respect to the shares issuable upon exercise of the Warrants and upon conversion under the Omnibus Agreement;

NOW,
THEREFORE, for and in consideration of the foregoing, and other good and valuable consideration, the sufficiency of which are hereby acknowledged, the parties hereto, intending to
be legally bound, do hereby agree as follows:

1.    Amendments.

(a)  The
"RECITALS" section of the Agreement is amended by deleting such section entirely and replacing it with the following:



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(b)  The
definition of Registrable Securities in Section 1(d) of the Agreement is amended by deleting such Section 1(d) entirely and replacing it with the following:



(c)  Section
2(a) of the Agreement is amended by deleting such Section 2(a) entirely and replacing it with the following:



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(d)  Section
2(b) of the Agreement is amended by deleting such Section 2(b) entirely and replacing it with the following:



(e)  Section
4(a) of the Agreement is amended by deleting such Section 4 entirely and replacing it with the following:



(f)    Section
5(a) of the Agreement is amended by deleting such Section 5(a) entirely and replacing it with the following:



(g)  Section
8(a) of the Agreement is amended by deleting such Section 8(a) entirely and replacing it with the following:



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(h)  Schedule
A to the Agreement is amended by deleting such Schedule A entirely and replacing it with Schedule A to this Amendment.

2.    No Other Changes.Except as explicitly amended by this Amendment, the terms, conditions, rights and
obligations under the Agreement shall remain in full force and effect.

3.    Counterparts.This Amendment may be executed by the parties hereto in any number of counterparts, each of
which shall be deemed an original, but all of which shall constitute one and the same agreement.

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IN WITNESS WHEREOF, the parties hereto have executed this Amendment as of the date set forth above.

|BRIGHAM EXPLORATION COMPANY                                                                                                                                |                                                              |                                                   
|---------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------ | --------------------------------------------------
|By:                                                                                                                                                        | /s/DAVID T. BRIGHAM                                          |                                                   
|Name:                                                                                                                                                      | David T. Brigham                                             |                                                   
|Title:                                                                                                                                                     | Executive Vice PresidentLand and Administration             |                                                   
|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/ROBERT CABESROBERT CABES
      Attorney-in-Fact
|DLJ ESC II, LP                                                                                                                                             |                                                              |                                                   
|By: DLJ LBO PLANS MANAGEMENT CORPORATION,
its general partner                                                                                             
|                                                                                                                                                           | By:                                                          | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact
|DLJ MERCHANT BANKING PARTNERS III, L.P.                                                                                                                   
|By:                                                                                                                                                        | DLJ MERCHANT BANKING III, INC.,
its Managing General Partner |                                                   
|                                                                                                                                                           | By:                                                          | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact

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|DLJ MERCHANT BANKING III, INC., AS ADVISORY
GENERAL PARTNER ON BEHALF OF
DLJ OFFSHORE PARTNERS III, C.V.                                                                                                                                 |     |                                                    
|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- | ---------------------------------------------------
|                                                                                                                                                                                                                                         | By: | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact 
|DLJ MERCHANT BANKING III, INC.,
AS ADVISORY GENERAL PARTNER ON BEHALF OF
DLJ OFFSHORE PARTNERS III-1, C.V. AND AS
ATTORNEY-IN-FACT FOR DLJ MERCHANT BANKING III, L.P.,
AS ASSOCIATE GENERAL PARTNER OF
DLJ OFFSHORE PARTNERS III-1, C.V.
|                                                                                                                                                                                                                                         | By: | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact 
|DLJ MERCHANT BANKING III, INC., AS ADVISORY
GENERAL PARTNER ON BEHALF OF
DLJ OFFSHORE PARTNERS III-2, C.V. AND
AS ATTORNEY-IN-FACT FOR DLJ MERCHANT BANKING III, L.P.,
AS ASSOCIATE GENERAL PARTNER OF
DLJ OFFSHORE PARTNERS III-2, C.V.
|                                                                                                                                                                                                                                         | By: | ROBERT CABES
      Attorney-in-Fact                
|DLJ MB PARTNERS III GmbH & CO. KG                                                                                                                                                                                                        |     |                                                    
|By: DLJ MERCHANT BANKING III, L.P.,
its Managing Limited Partner                                                                                                                                                                        
|                                                                                                                                                                                                                                         | By: | DLJ MERCHANT BANKING III, INC.,
its General Partner
|                                                                                                                                                                                                                                         | By: | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact 

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|MILLENNIUM PARTNERS II, L.P.                                                                                |     |                                                   
|----------------------------------------------------------------------------------------------------------- | --- | --------------------------------------------------
|By: DLJ MERCHANT BANKING III, INC.,
its Managing General Partner                                           
|                                                                                                            | By: | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact
|MBP III PLAN INVESTORS, L.P.                                                                                |     |                                                   
|By: DLJ LBO PLANS MANAGEMENT CORPORATION,
its Managing General Partner                                      |     |                                                   
|                                                                                                            | By: | /s/ROBERT CABESROBERT CABES
      Attorney-in-Fact
|ADDRESSES FOR NOTICES TO CSFB ENTITIES:                                                                    
|Global Energy Partners
1100 Louisiana Street
Houston, Texas 77002
Fax: 713-890-1429
Attn: Steven A. Webster
|and                                                                                                        
|CSFB Private Equity
11 Madison Avenue
New York, New York 10010
Fax: 212-448-3502
Attn: Dorian S. Forshner  

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Schedule A  to Registration Rights Agreement

|                                                                                                                                                                                                                                         |  | Number of Shares
|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |  | ----------------
|
|DLJ Merchant Banking Partners III, L.P.                                                                                                                                                                                                  |  | 4,681,393       
|DLJ MB Funding III, Inc                                                                                                                                                                                                                  |  | 125,748         
|DLJ ESC II, LP                                                                                                                                                                                                                           |  | 1,409,326       
|DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners III, C.V.                                                                                                                                 |  | 255,680         
|DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners III-1, C.V. and as attorney-in-fact for DLJ Merchant Banking III, L.P., as Associate General Partner of DLJ Offshore Partners III-1, C.V. |  | 85,360          
|DLJ Merchant Banking III, Inc., as Advisory General Partner on behalf of DLJ Offshore Partners III-2, C.V. and as attorney-in-fact for DLJ Merchant Banking III, L.P., as Associate General Partner of DLJ Offshore Partners III-2, C.V. |  | 60,807          
|DLJ MB Partners III GmbH & Co. KG                                                                                                                                                                                                        |  | 40,340          
|Millennium Partners II, L.P.                                                                                                                                                                                                             |  | 8,013           
|MBP III Plan Investors, L.P.                                                                                                                                                                                                             |  | -0-             
|Total                                                                                                                                                                                                                                    |  | 6,666,667       

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