The ergodic control problem for a non-degenerate diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White's `relative value iteration' algorithm for solving the ergodic dynamic programming equation for the finite state finite action case. Its convergence to the solution of the HJB equation is established using the theory of monotone dynamical systems and also, alternatively, by using the theory of reverse martingales.