For a nonlinear optimal control problem with state constraints, we give conditions under which the optimal control depends Lipschitz continuously in the L-2 norm on a parameter. These conditions involve smoothness of the problem data, uniform independence of active constraint gradients, and a coercivity condition for the integral functional. Under these same conditions, we obtain a new nonoptimal stability result for the optimal control in the L-infinity norm. And under an additional assumption concerning the regularity of the state constraints, a new tight L-infinity estimate is obtained. Our approach is based on an abstract implicit function theorem in nonlinear spaces.