The problem of asymptotic stabilization for a class of nonholonomic systems is studied and solved by means of a hybrid control law which makes use of a (deterministic) finite state machine. It is shown that, by using a simple switching control scheme, the origin is a globally asymptotically stable equilibrium in the sense of Lyapunov. The control law can take into account the presence of input saturation. Simulation results are reported showing the performance of the proposed control scheme.