  We present results of an extensive observing campaign of the short duration,
hard spectrum gamma-ray burst (GRB) 070724A, aimed at detecting the
radioactively-powered emission that might follow from a binary merger or
collapse involving compact objects. Our multi-band observations span the range
in time over which this so-called Li-Paczynski mini-supernova could be active,
beginning within 3 hours of the GRB trigger, and represent some of the deepest
and most comprehensive searches for such emission. We find no evidence for such
activity and place limits on the abundances and the lifetimes of the possible
radioactive nuclides that could form in the rapid decompression of
nuclear-density matter. Furthermore, our limits are significantly fainter than
the peak magnitude of any previously detected broad-lined Type Ic supernova
(SN) associated with other GRBs, effectively ruling out a long GRB-like SN for
with this event. Given the unambiguous redshift of the host galaxy (z=0.456),
GRB 070724A represents one of a small, but growing, number of short-hard GRBs
for which firm physical/restframe quantities currently exist. The host of GRB
070724A is a moderately star-forming galaxy with an older stellar population
component and a relatively high metallicity of 12+log(O/H)_KD02=9.1. We find no
significant evidence for large amounts of extinction along the line of sight
that could mask the presence of a SN explosion and estimate a small probability
for chance alignment with the putative host. We discuss how our derived
constraints fit into the evolving picture of short-hard GRBs, their potential
progenitors, and the host environments in which they are thought to be
produced.
