AMENDMENT NUMBER TWO TO EXCLUSIVE LICENSE AGREEMENT

Exhibit 10.6.2

A MENDMENT N UMBER T WO

TO

E XCLUSIVE L ICENSE A GREEMENT

BETWEEN

T HE R EGENTS OF THE U NIVERSITY OF C ALIFORNIA AND N EUROGES X, I NC .

This Amendment Number Two is made effective December 2, 2003 (the “Amendment Date”) between THE REGENTS OF THE UNIVERSITY OF CALIFORNIA, a California corporation having its statewide administrative offices at 1111 Franklin Street, Oakland, California 94607-5200, (“The Regents”), and acting through its Office of Technology Management, University of California San Francisco, 185 Berry Street, Suite 4603, San Francisco, CA 94107 (“UCSF”) and NEUROGESX, INC., a California corporation having a principal place of business at 981F Industrial Boulevard, San Carlos, California, (the “Licensee”) (together “the Parties”). The Parties hereby amend that certain Exclusive License Agreement, dated as of November 1, 2000 (the “Agreement”) and Amendment Number One dated as of November 1, 2001 (“Amendment One”).

WITNESSETH

WHEREAS, The Regents and Licensee entered into the Agreement and Amendment One;

NOW, THEREFORE,