  [Abridged] We present the results of an extensive Hubble Space Telescope
(HST) imaging study of ~100 Swift long-duration gamma-ray bursts (LGRBs)
spanning 0.03 < z < 9.4 using relative astrometry from afterglow observations
to locate the bursts within their host galaxies. We measure the distribution of
LGRB offsets from their host centers, as well as their relation to the
underlying host light distribution. We find that the host-normalized offsets
(R/R_h) of LGRBs are more centrally concentrated than expected for an
exponential disk profile, with a median of 0.67, and in particular they are
more concentrated than the underlying surface brightness profiles of their host
galaxies. The distribution of offsets is inconsistent with the distribution for
Type II supernovae (SNe) but consistent with the distribution for Type Ib/c
SNe. The fractional flux distribution, with a median of 0.75, indicates that
LGRBs prefer some of the brightest locations in their host galaxies but are not
as strongly correlated as previous studies indicated. More importantly, we find
a clear correlation between the offset and fractional flux, where bursts at
R/R_h < 0.5 exclusively occur at fractional fluxes > 0.6 while bursts at R/R_h
> 0.5 uniformly trace the light of their hosts. This indicates that the spatial
correlation of LGRB locations with bright star forming regions seen in the full
sample is dominated by the contribution from bursts at small offset and that
LGRBs in the outer parts of galaxies show no preference for unusually bright
star forming regions. We conclude that LGRBs strongly prefer the bright, inner
regions of their hosts indicating that the star formation taking place there is
more favorable for LGRB progenitor production. This indicates that another
environmental factor beyond metallicity, such as binary interactions or IMF
differences, may be operating in the central regions of LGRB hosts.
