The stability of optimal solutions of stochastic programming problems with respect to the underlying probability distribution is studied by means of results for sensitvity analysis of nonlinear programs. Gâteaux derivatives of optimal solutions are obtained under relatively weak assumptions about the ``true'' stochastic program under which the ``true'' optimal solution is an isolated minimizer but the set of the corresponding Lagrange multipliers need not be singleton. Under additional assumptions, these results are used to characterize an asymptotic distribution of optimal solutions of the stochastic program in which the ``true'' probability distribution is replaced by the empirical one.