Rescaled contact processes converge to super-Brownian motion in two or more dimensions

We show that in dimensions two or more a sequence of long range contact pro cesses suitably rescaled in space and time converges to a super-Brownian mo tion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnit ude estimates of how close the critical value is to I with sharp asymptotic s.