Exhibit 10.42

AGREEMENT RELATING TO VOTING OF SHARES

This AGREEMENT RELATING TO VOTING OF SHARES (this “Agreement”), is entered into
July 31, 2008 by and between DLJ Merchant Banking Partners III, L.P., DLJ
Offshore Partners III, C.V., DLJ Offshore Partners III-1, C.V., DLJ Offshore
Partners III-2, C.V., DLJ MB PartnersIII GmbH & Co. KG, Millennium Partners II,
L.P., MBP III Plan Investors, L.P., DLJ ESC II, L.P. and DLJMB Funding III, Inc.
(collectively, the “CS Entities”), and Brigham Exploration Company, a Delaware
corporation (the “Company”).

WHEREAS, the CS Entities own more than 9.99% of the common stock of the Company
(the “Common Stock”);

WHEREAS, the CS Entities have determined that it is no longer necessary to
retain their right to nominate a representative to the Board of Directors of the
Company and are entering into an Amendment to Securities Purchase Agreement to
relinquish that right;

WHEREAS, the CS Entities have also determined to limit the number of shares of
Common Stock that they will vote at any stockholders’ meeting of the Company;

NOW, THEREFORE, for good and valuable consideration the receipt and sufficiency
of which are hereby acknowledged and confessed, and in consideration of the
premises and the mutual covenants and agreements herein contained, and intending
to be legally bound hereby, the Company and the CS Entities hereby agree as
follows:

1. Agreement to Limit Voting of Shares. Regardless of the number of shares of
Common Stock owned by them at any time, the CS Entities will not vote more than
9.99% of the outstanding shares of Common Stock whether at a meeting of the
stockholders of the Company or otherwise. Without limiting the foregoing, all of
the shares of Common Stock owned by the CS Entities will retain their character
as outstanding shares of Common Stock entitled to vote on any matter submitted
to the stockholders of the Company. For purposes of this Section 1, shares of
Common Stock that are present at a meeting and that abstain on any matter will
not be considered to have been voted on that matter.

2. No Company Liability. The CS Entities undertake to ensure that the
requirements of Section 1 are fulfilled. The Company will not be liable to the
CS Entities or to any other person for any claims arising from the Company’s
good faith efforts to enforce the requirements of Section 1 or for any claims
arising from the Company’s failure to enforce the requirements of Section 1.

3. Indemnification. The CS Entities, severally and not jointly, shall indemnify
and hold the Company and each of its directors, officers, employees and agents
(each a “Company Party”), harmless from any and all losses, liabilities,
obligations, claims, contingencies, damages, costs, and expenses (including all
court costs and reasonable attorneys’ fees) that any Company Party may suffer or
incur as a result of, arising out of or in connection with this Agreement or
anything done or omitted to be done by such Company Party in connection with
this Agreement, including the negligence or alleged negligence of any Company
Party, but excluding the gross negligence, fraud or willful misconduct of any
Company Party.

 

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4. Amendment; Waiver. This Agreement may not be amended except by an instrument
in writing signed by or on behalf of all the parties hereto. No failure or delay
by a party hereto in exercising any right, power, or privilege hereunder shall
operate as a waiver thereof nor shall any single or partial exercise thereof
preclude any other or further exercise thereof or the exercise of any other
right, power, or privilege. The provisions of this Agreement may not be waived
except by an instrument in writing signed by or on behalf of the party against
whom such waiver is sought to be enforced.

5. No Third Party Beneficiaries. Nothing in this Agreement, express or implied,
is intended to or shall confer upon any person other than the parties hereto,
and their respective successors and permitted assigns, any rights, benefits or
remedies of any nature whatsoever under or by reason of this Agreement.

6. Counterparts. This Agreement 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. Each counterpart may consist
of a number of copies hereof, each signed by less than all, but together signed
by all of the parties hereto.

IN WITNESS WHEREOF, the parties have executed this Agreement or caused this
Agreement to be executed by their duly authorized representatives, all as of the
day and year first above written.

THE COMPANY:

BRIGHAM EXPLORATION COMPANY

By: /s/ Eugene B. Shepherd, Jr.                 
Name: Eugene B. Shepherd, Jr.
Title: Chief Financial Officer

CS ENTITIES

DLJ MERCHANT BANKING PARTNERS III, L.P.
By: DLJ Merchant Banking III, Inc.,
       Its Managing General Partner

By: /s/ William Spiro                               
Name: William Spiro
Title: Vice President

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DLJ MERCHANT BANKING III, INC., as Advisory General Partner on behalf of DLJ
OFFSHORE PARTNERS III, C.V.

By: /s/ William Spiro                                     
Name: William Spiro
Title: Vice President

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 DLJMB III (Bermuda),
L.P., as Associate General Partner of DLJ OFFSHORE PARTNERS III-1, C.V.

By: /s/ William Spiro                                    
Name: William Spiro
Title: Vice President

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 DLJMB III (Bermuda),
L.P., as Associate General Partner of DLJ OFFSHORE PARTNERS III-2, C.V.

By: /s/ William Spiro                                  
Name: William Spiro
Title: Vice President

DLJ MERCHANT BANKING III, INC., as General Partner of DLJ Merchant Banking III,
L.P. and as attorney-in-fact for DLJ Merchant Banking III, L.P. as Managing
Limited Partner for and on behalf of DLJ MB PARTNERSIII GmbH & Co. KG

By: /s/ William Spiro                                    
Name: William Spiro
Title: Vice President

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DLJ MB GmbH, as General Partner for and on behalf of DLJ MB PARTNERSIII GmbH &
Co. KG

By: /s/ Edward Nadel                                   
Name: Edward Nadel
Title: Director

MILLENNIUM PARTNERS II,, INC.L.P.
By: DLJ Merchant Banking III, Inc.,
       Its Managing General Partner

By: /s/ William Spiro                                  
Name: William Spiro
Title: Vice President

MBP III PLAN INVESTORS, L.P.
By: DLJ LBO Plans Management Corporation II,
       Its General Partner

By: /s/ William Spiro                                 
Name: William Spiro
Title: Vice President

DLJ ESC II, INC.L.P.
By: DLJ LBO Plans Management Corporation,
       Its General Partner

By: /s/ William Spiro                                
Name: William Spiro
Title: Vice President

DLJMB FUNDING III, INC.

By: /s/ William Spiro                                
Name: William Spiro
Title: Vice President

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