  This article deals with the limit distribution for a stochastic differential
equation driven by a non-symmetric cylindrical $\alpha$-stable process. Under
suitable conditions, it is proved that the solution of this equation converges
weakly to that of a stochastic differential equation driven by a Brownian
motion in the Skorohod space as $\alpha\rightarrow2$. Also, the rate of weak
convergence, which depends on $ 2-\alpha$, for the solution towards the
solution of the limit equation is obtained. For illustration, the results are
applied to a simple one-dimensional stochastic differential equation, which
implies the rate of weak convergence is optimal.
