  We give criteria for ergodicity, transience and null recurrence for the
random walk in random environment on {0,1,2,...}, with reflection at the
origin, where the random environment is subject to a vanishing perturbation.
Our results complement existing criteria for random walks in random
environments and for Markov chains with asymptotically zero drift, and are
significantly different to these previously studied cases. Our method is based
on a martingale technique - the method of Lyapunov functions.
