We prove a Tauberian theorem for nonexpansive operators, and apply it to the model of zero-sum stochastic game. Under mild assumptions, we prove that the value of the λ-discounted game v λ converges uniformly when λ goes to 0 if and only if the value of the n-stage game v n converges uniformly when n goes to infinity. This generalizes the Tauberian theorem of Lehrer and Sorin [6] to the two-player zero-sum case. We also provide the first example of a stochastic game with public signals on the state and perfect observation of actions, with finite state space, signal sets and action sets, in which for some initial state k 1 known by both players, (v λ (k 1 )) and (v n (k 1 )) converge to distinct limits.