Recent developments in numerical relativity have made it possible to reliably follow the coalescence of two black holes from near the innermost stable circular orbit to final ringdown. This opens up a wide variety of exciting astrophysical applications of these simulations. Chief among these is the net kick received when two unequal mass or spinning black holes merge. The magnitude of this kick has bearing on the production and growth of supermassive black holes during the epoch of structure formation, and on the retention of black holes in stellar clusters. Here we report the first accurate numerical calculation of this kick, for two nonspinning black holes in a 1.5: 1 mass ratio, which is expected on the basis of analytic considerations to give a significant fraction of the maximum possible recoil. We have performed multiple runs with different initial separations, orbital angular momenta, resolutions, extraction radii, and gauges. The full range of our kick speeds is 86-116 km s super(-1), and the most reliable runs give kicks between 86 and 97 km s super(-1). This is intermediate between the estimates from two recent post-Newtonian analyses and suggests that at redshifts z 10, halos with masses 10 super(9) M sub( )will have difficulty retaining coalesced black holes after major mergers.