  We present the discovery of the optical afterglow and early-type host galaxy
of the short-duration GRB 100117A. The faint afterglow is detected 8.3 hr after
the burst with r_AB = 25.46 +/- 0.20 mag. Follow-up optical and near-IR
observations uncover a coincident compact red galaxy, identified as an
early-type galaxy at a photometric redshift of z~0.6-0.9 (2-sigma) with a mass
of 3x10^10 M_Sun, an age of ~1 Gyr, and a luminosity of L_B~0.5L_star.
Spectroscopic observations of the host reveal a notable break corresponding to
the Balmer 4000-Angstrom break at z~0.9, and stellar population spectral
evolution template fits indicate z~0.915, which we adopt as the redshift of the
host, with stellar population ages of ~1-3 Gyr. From a possible weak detection
of [OII]-3727 emission at z=0.915 we infer an upper bound on the star formation
rate of ~0.1 M_Sun per yr, leading to a specific star formation rate of <0.004
per Gyr. Thus, GRB 100117A is only the second short burst to date with a secure
early-type host (the other being GRB 050724 at z=0.257) and it has one of the
highest short GRB redshifts. The offset between the host center and the burst
position, 470 +/- 310 pc, is the smallest to date. Combined with the old
stellar population age, this indicates that the burst likely originated from a
progenitor with no significant kick velocity. However, from the brightness of
the optical afterglow we infer a relatively low density of n~3x10^-4 cm^-3 when
epsilon_e and epsilon_B = 0.1. The combination of an optically faint afterglow
and host suggest that previous such events may have been missed, thereby
potentially biasing the known short GRB host population against z>1 early-type
hosts.
