Exhibit 32

CERTIFICATIONS PURSUANT TO 18 U.S.C. SECTION 1350

               In connection with the Quarterly Report of ConocoPhillips (the company) on Form 10-Q for the period ended September 30, 2008, as filed with the U.S. Securities and Exchange Commission on the date hereof (the Report), each of the undersigned hereby certifies, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that to their knowledge:

(1)	The Report fully complies with the requirements of Sections 13(a) or 15(d) of the Securities Exchange Act of 1934; and
(2)	The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the company.

Date: October 29, 2008

/s/ James J. Mulva  

James J. Mulva 

Chairman and Chief Executive Officer 

/s/ Sigmund L. Cornelius  

Sigmund L. Cornelius 

Senior Vice President, Finance, and
Chief Financial Officer