With the increasing reliance on game theory as a foundation for auctions and electronic commerce, efficient algorithms for computing equilibria in multiplayer general-sum games are of great theoretical and practical interest. The computational complexity of finding a Nash equilibrium for a one-shot bimatrix game is a well known open problem. This paper treats a closely related problem, that of finding a Nash equilibrium for an average-payoff phrepeated bimatrix game, and presents a polynomial-time algorithm. Our approach draws on the "folk theorem" from game theory and shows how finite-state equilibrium strategies can be found efficiently and expressed succinctly.