This paper deals with Borel state and action spaces zero-sum semi-Markov games under the expected long run average payo criterion. The transition probabilities are assumed to satisfy some generalized geometric ergodicity conditions. The main result states that the optimality equation has a solution, which is approximated by the solutions of some epsilon-perturbed semi-Markov games. As a corollary, the existence of value and average optimal strategies for the players is established.