  The known host galaxies of short-hard gamma-ray bursts (GRBs) to date are
characterized by low to moderate star-formation rates and a broad range of
stellar masses. In this paper, we positionally associate the recent
unambiguously short-hard Swift GRB 100206A with a disk galaxy at redshift
z=0.4068 that is rapidly forming stars at a rate of ~30 M_sun/yr, almost an
order of magnitude higher than any previously identified short GRB host. Using
photometry from Gemini, Keck, PAIRITEL, and WISE, we show that the galaxy is
very red (g-K = 4.3 AB mag), heavily obscured (A_V ~ 2 mag), and has the
highest metallicity of any GRB host to date (12 + log[O/H]_KD02 = 9.2): it is a
classical luminous infrared galaxy (LIRG), with L_IR ~ 4 x 10^11 L_sun. While
these properties could be interpreted to support an association of this GRB
with very recent star formation, modeling of the broadband spectral energy
distribution also indicates that a substantial stellar mass of mostly older
stars is present. The current specific star-formation rate is modest (specific
SFR ~ 0.5 Gyr^-1), the current star-formation rate is not substantially
elevated above its long-term average, and the host morphology shows no sign of
recent merger activity. Our observations are therefore equally consistent with
an older progenitor, similar to what is inferred for other short-hard GRBs.
Given the precedent established by previous short GRB hosts and the significant
fraction of the Universe's stellar mass in LIRG-like systems at z >~0.3, an
older progenitor represents the most likely origin of this event.
