Despite a rich phenomenology, gamma-ray bursts (GRBs) are divided into two classes based on their duration and spectral hardness--the long-soft and the short-hard bursts. The discovery of afterglow emission from long GRBs was a watershed event, pinpointing their origin to star-forming galaxies, and hence the death of massive stars, and indicating an energy release of about 10(51) erg. While theoretical arguments suggest that short GRBs are produced in the coalescence of binary compact objects (neutron stars or black holes), the progenitors, energetics and environments of these events remain elusive despite recent localizations. Here we report the discovery of the first radio afterglow from the short burst GRB 050724, which unambiguously associates it with an elliptical galaxy at a redshift z = 0.257. We show that the burst is powered by the same relativistic fireball mechanism as long GRBs, with the ejecta possibly collimated in jets, but that the total energy release is 10-1,000 times smaller. More importantly, the nature of the host galaxy demonstrates that short GRBs arise from an old (> 1 Gyr) stellar population, strengthening earlier suggestions and providing support for coalescing compact object binaries as the progenitors.