The final chapter in the long-standing mystery of the γ-ray bursts (GRBs) centres on the origin of the short-hard class of bursts, which are suspected on theoretical grounds to result from the coalescence of neutron-star or black-hole binary systems. Numerous searches for the afterglows of short-hard bursts have been made, galvanized by the revolution in our understanding of long-duration GRBs that followed the discovery in 1997 of their broadband (X-ray, optical and radio) afterglow emission. Here we present the discovery of the X-ray afterglow of a short-hard burst, GRB 050709, whose accurate position allows us to associate it unambiguously with a star-forming galaxy at redshift z = 0.160, and whose optical lightcurve definitively excludes a supernova association. Together with results from three other recent short-hard bursts, this suggests that short-hard bursts release much less energy than the long-duration GRBs. Models requiring young stellar populations, such as magnetars and collapsars, are ruled out, while coalescing degenerate binaries remain the most promising progenitor candidates.