  We report optical and near-infrared broad band observations of the
short-duration GRB 050724 host galaxy, used to construct its spectral energy
distribution (SED). Unlike the hosts of long-duration gamma-ray bursts (GRBs),
which show younger stellar populations, the SED of the GRB 050724 host galaxy
is optimally fitted with a synthetic elliptical galaxy template based on an
evolved stellar population (age ~2.6 Gyr). The SED of the host is difficult to
reproduce with non-evolving metallicity templates. In contrast, if the short
GRB host galaxy metallicity enrichment is considered, the synthetic templates
fit the observed SED satisfactorily. The internal host extinction is low (A_v
\~< 0.4 mag) so it cannot explain the faintness of the afterglow. This short
GRB host galaxy is more massive (~5x10^10 Mo) and luminous (~1.1 L*) than most
of the long-duration GRB hosts. A statistical comparison based on the ages of
short- and long-duration GRB host galaxies strongly suggests that
short-duration GRB hosts contain, on average, older progenitors. These findings
support a different origin for short- and long-duration GRBs.
