Gamma-ray bursts (GRBs) fall into two classes: short-hard and long-soft bursts. The latter are now known to have X-ray and optical afterglows, to occur at cosmological distances in star-forming galaxies, and to be associated with the explosion of massive stars. In contrast, the distance scale, the energy scale and the progenitors of the short bursts have remained a mystery. Here we report the discovery of a short-hard burst whose accurate localization has led to follow-up observations that have identified the X-ray afterglow and (for the first time) the optical afterglow of a short-hard burst; this in turn led to the identification of the host galaxy of the burst as a late-type galaxy at z = 0.16 (ref. 10). These results show that at least some short-hard bursts occur at cosmological distances in the outskirts of galaxies, and are likely to be caused by the merging of compact binaries.