In this paper we consider two-person zero-sum stochastic games with the average reward criterion and Borel state and action spaces. A geometric drift condition is assumed. We show that the optimality (Shapley) equation has a unique solution if the transition probability function is weakly continuous, the stage reward is lower semicontinuous, and the set-valued mappings of admissible actions satisfy some semicontinuity assumptions. Furthermore, the minimizing player has an optimal stationary strategy and the maximizing player has an epsilon-optimal stationary strategy for every epsilon > 0.