  In the present paper we study the convergence of the solution of the two
dimensional (2-D) stochastic Leray-$\alpha$ model to the solution of the 2-D
stochastic Navier-Stokes equations. We are mainly interested in the rate of
convergence, as $\alpha$ tends to 0, of the error function which is the
difference between the two solutions in an appropriate topology. We show that
when properly localized the error function converges in mean square as
$\alpha\to 0$ and the convergence is of order $O(\alpha)$. We also prove that
the error converges in probability to zero with order at most $O(\alpha)$.
