AMENDMENTNO. 1 TO RIGHTSAGREEMENT

EXHIBIT 4.1

AMENDMENT NO. 1

TO

RIGHTS AGREEMENT

This Amendment No. 1 to Rights Agreement is entered into as of September 23, 2004, between Endocardial Solutions, Inc., a Delaware corporation (the “Company”), and Wells Fargo Bank, National Association (f/k/a Norwest Bank Minnesota, National Association), a national banking association (the “Rights Agent”).

WHEREAS, the Company and the Rights Agent are parties to a Rights Agreement dated as of August 25, 1999 (the “Rights Agreement”);

WHEREAS, St. Jude Medical, Inc., a Minnesota corporation (“Buyer”), Dragonfly Merger Corp., a Delaware corporation (“Buyer Subsidiary”), and the Company propose to enter into an Agreement and Plan of Merger (the “Merger Agreement”), pursuant to which Buyer Subsidiary will merge with and into the Company (the “Merger”);

WHEREAS, the Board of Directors of the Company has approved the Merger Agreement; and

WHEREAS, the Board of Directors of the Company has determined that an amendment to the Rights Agreement pursuant to Section 27 of the Rights Agreement is desirable in connection with the foregoing and the Company and the Rights Agent desire to evidence such amendment in writing.

NOW, THEREFORE, in consideration of the foregoing and the mutual agreements set forth herein, the parties hereto agree as follows:

(a)           Amendment to Section 1.  Section 1 of the Rights Agreement is amended to add the following sentence at the end of the definition of “Acquiring Person”:

“Notwithstanding anything in this Agreement to the contrary, neither Buyer nor Buyer Subsidiary shall be deemed an “Acquiring Person” as the result of (i) the consummation of the Merger, (ii) the execution or delivery of the Merger Agreement or (iii) the consummation of the transactions contemplated by the Merger Agreement.”

Section 1 of the Rights Agreement is further amended to add the following sentence at the end of the definition of “Beneficial Owner”:

“Notwithstanding anything in this Agreement to the contrary, neither Buyer nor Buyer Subsidiary shall be deemed to beneficially own any securities of the Company that Buyer or Buyer Subsidiary would otherwise beneficially own as a result of (i) the consummation of the Merger, (ii) the execution or delivery of the Merger Agreement or (iii) the consummation of the other transactions contemplated in the Merger Agreement.”

Section 1 of the Rights Agreement is further amended to add the following sentence at the end of the definition of “Shares Acquisition Date”:

“Notwithstanding anything in this Agreement to the contrary, no “Shares Acquisition Date” shall be deemed to occur as a result of (i) the consummation of the Merger, (ii) the execution or delivery of the Merger Agreement or (iii) the consummation of the other transactions contemplated in the Merger Agreement.”

(b)           Additional Amendment to Section 1.  Section 1 of the Rights Agreement is further amended to add the following paragraphs at the end thereof:

“Buyer” shall mean St. Jude Medical, Inc., a Minnesota corporation, and its Affiliates and Associates.

“Buyer Subsidiary” shall mean Dragonfly Merger Corp., a wholly owned subsidiary of Buyer, and its Affiliates and Associates.

“Merger” shall mean the merger of Buyer Subsidiary with and into the Company pursuant to the Merger Agreement.

“Merger Agreement” shall mean that certain Agreement and Plan of Merger dated as of September 23, 2004 by and among the Company, Buyer and Buyer Subsidiary.

(c)           Amendment to Section 3(a).  Section 3(a) of the Rights Agreement is amended to add the following sentence at the end thereof:

“Notwithstanding anything in this Agreement to the contrary, a Distribution Date shall not be deemed to have occurred as the result of (i) the consummation of the Merger, (ii) the execution or delivery of the Merger Agreement or (iii) the consummation of the other transactions contemplated in the Merger Agreement.”

(d)           Effectiveness.  This Amendment shall be deemed effective immediately prior to the execution and delivery of the Merger Agreement.  Except as amended hereby, the Rights Agreement shall remain in full force and effect and shall be otherwise unaffected hereby.

(e)           Termination of Rights Agreement.  The Rights Agreement shall terminate immediately upon effectiveness of the Merger.

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IN WITNESS WHEREOF, the parties hereto have caused this Amendment to be duly executed, all as of the date and year first above written.

Signature Page to

Amendment No. 1 to Rights Agreement

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