This paper studies the policy iteration algorithm (PIA) for zero-sum stochastic differential games with the basic long-run average criterion, as well as with its more selective version, the so-called bias criterion. The system is assumed to be a nondegenerate diffusion. We use Lyapunov-like stability conditions that ensure the existence and boundedness of the solution to certain Poisson equation. We also ensure the convergence of a sequence of such solutions, of the corresponding sequence of policies, and, ultimately, of the PIA.