In this paper the problem of the existence and determining stationary Nash equilibria in a single-controller average stochastic game is considered. The set of states and the set of actions in the game are assumed to be finite. We show that all stationary equilibria for such a game can be obtained from an auxiliary noncooperative static game in normal form where the payoffs are quasi-monotonic (quasi-convex and quasi-concave) with respect to the corresponding strategies of the players and graph-continuous in the sense of Dasgupta and Maskin. Based on this we present a proof of the existence of stationary equilibria in a single-controller average stochastic game and propose an approach for determining the optimal stationary strategies of the players.