  The localization of the short-duration, hard-spectrum GRB 050509b was a
watershed event. Thanks to the nearly immediate relay of the GRB position by
Swift, we began imaging the GRB field 8 minutes after the burst and continued
for the following 8 days. No convincing optical/infrared candidate afterglow or
supernova was found for the object. We present a re-analysis of the XRT
afterglow and find an absolute position that is ~4" to the west of the XRT
position reported previously. Close to this position is a bright elliptical
galaxy with redshift z=0.2248, about 1' from the center of a rich cluster of
galaxies. Based on positional coincidences, the GRB and the bright elliptical
are likely to be physically related. We thus have discovered evidence that at
least some short-duration, hard-spectra GRBs arise at cosmological distances.
However, while GRB 050509b was underluminous compared to long-duration GRBs, we
demonstrate that the ratio of the blast-wave energy to the gamma-ray energy is
consistent with that of long-duration GRBs. Based on this analysis, on the
location of the GRB (40 +- 13 kpc from a bright galaxy), on the galaxy type
(elliptical), and the lack of a coincident supernova, we suggest that there is
now observational consistency with the hypothesis that short-hard bursts arise
during the merger of a compact binary. We limit the properties of a
Li-Paczynski ''mini-supernova.'' Other progenitor models are still viable, and
additional rapidly localized bursts from the Swift mission will undoubtedly
help to further clarify the progenitor picture. (abridged)
