[Letterhead of Goldman, Sachs & Co.]

Exhibit 10.5

Exhibit 10.5

[Letterhead of Goldman, Sachs & Co.]

PERSONAL AND CONFIDENTIAL

April 15, 2005

Marshall Carter

Chairman

New York Stock Exchange, Inc.

11 Wall Street

New York, NY 10005

Dear Mr. Carter:

We refer to the letter (the “Authorization Letter”), dated February 10, 2005, between the New York Stock Exchange, Inc. (the “Company”) and Goldman, Sachs & Co. (“Goldman Sachs”) pursuant to which Goldman Sachs is authorized by the Company to facilitate exploring a potential strategic transaction between the Company and Archipelago Holdings, Inc. (“Archipelago”). We note that pursuant to the tenth paragraph of the Authorization Letter, our services automatically terminated 30 days from the date of the Authorization Letter (the “Automatic Termination Date”). The Company and Goldman Sachs hereby confirm that Goldman Sachs was reauthorized as of the Automatic Termination Date to continue our services pursuant to the terms of the Authorization Letter and the Company and Goldman Sachs agree that the Authorization Letter (including Annex A thereto) was, as of the Automatic Termination Date (and continues to be), in full force and effect. Goldman Sachs and the Company further agree that our services will automatically terminate on the date thirty days after the date of this letter unless we shall agree in writing to extend them for a specified period; provided that if a Transaction (as defined in the Authorization Letter) is announced prior to expiration of the thirty-day period from the date of this letter, our services will not automatically terminate on such thirtieth day and such automatic termination date shall be extended until consummation of a Transaction.

In addition, the Company and Goldman Sachs agree that the Transaction Fee referred to in the fourth paragraph of the Authorization Letter shall be $3.5 million.

Except as set forth herein, the Company and Goldman Sachs agree that all of the other terms and obligations in the Authorization Letter shall continue in full force and effect without modification.

New York Stock Exchange, Inc.

April 15, 2005

Page Two

Please confirm that the foregoing is in accordance with your understanding by signing and returning to us the enclosed copy of this letter, which shall become a binding agreement upon our receipt.