Let

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment in the case that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer [13] for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with nite state space or with transition kernel dominated above and below by a probability measure. MSC2000: primary 60K37, 60F05; secondary 60J05, 60J80