We consider stochastic nonaffine nonlinear control systems x(t) = x(o) + integral(o)(t) f(x(s), u) ds + integral(o)(t) g(x(s), u) dw(s) (written in the sense of Ito), w being a standard Wiener process, for which we give a sufficient condition for global stabilization by a bounded smooth state feedback that is explicitly given. This condition generalizes the well-known Jurdjevic-Quinn result for deterministic affine control systems.