The localization of the short-duration, hard-spectrum gamma-ray burst GRB 050509b by the Swift satellite was a watershed event. We report the discovery of the probable host galaxy, a bright elliptical galaxy at z=0.2248. This is the first known redshift and host of a short-hard GRB and shows that at least some short- hard GRBs are cosmological in origin. We began imaging the GRB field 8 minutes after the burst and continued for 8 days. We present a reanalysis of the XRT afterglow and report the absolute position of the GRB. Based on positional coincidences, the GRB and the elliptical are likely to be physically related, unlike any known connection between a long-duration GRB and an early-type galaxy. Similarly unique, GRB 050509b likely also originated from within a rich cluster of galaxies with detectable diffuse X-ray emission. We demonstrate that while the burst was underluminous, the ratio of the blast wave energy to the γ-ray energy is consistent with that of long-duration GRBs. Based on this analysis, on the location of the GRB (40+/-13 kpc from the putative host), on the galaxy type (elliptical), and the lack of a coincident supernova, we suggest that there is now observational support for the hypothesis that short-hard bursts arise during the merger of a compact binary. We limit the properties of any Li-Paczyʼnski ``minisupernova’’ that is predicted to arise on åisebox-0.5ex 1 day timescales. Other progenitor models are still viable, and new Swift bursts will undoubtedly help to further clarify the progenitor picture.