  We present a detailed investigation of Hubble Space Telescope (HST)
rest-frame UV/optical observations of 22 short gamma-ray burst (GRB) host
galaxies and sub-galactic environments. Utilizing the high angular resolution
and depth of HST, we characterize the host galaxy morphologies, measure precise
projected physical and host-normalized offsets between the bursts and host
centers, and calculate the locations of the bursts with respect to their host
light distributions (rest-frame UV and optical). We calculate a median short
GRB projected physical offset of 4.5 kpc, about 3.5 times larger than that for
long GRBs, and find that ~25% of short GRBs have offsets of >10 kpc. When
compared to their host sizes, the median offset is 1.5 half-light radii (r_e),
about 1.5 times larger than the values for long GRBs, core-collapse supernovae,
and Type Ia supernovae. In addition, ~20% of short GRBs having offsets of >5
r_e, and only ~25% are located within 1 r_e. We further find that short GRBs
severely under-represent their hosts' rest-frame optical and UV light, with
~30-45% of the bursts located in regions of their host galaxies that have no
detectable stellar light, and ~55% in the regions with no UV light. Therefore,
short GRBs do not occur in regions of star formation or even stellar mass. This
demonstrates that the progenitor systems of short GRBs must migrate from their
birth sites to their eventual explosion sites, a signature of kicks in compact
object binary systems. Utilizing the full sample of offsets, we estimate natal
kick velocities of ~20-140 km s^-1. These independent lines of evidence provide
the strongest support to date that short GRBs result from the merger of compact
object binaries (NS-NS/NS-BH).
