  We present a comprehensive study to measure the locations of gamma-ray bursts
(GRBs) relative to their host galaxies. In total, we find the offsets of 20
long-duration GRBs from their apparent host galaxy centers utilizing
ground-based images from Palomar and Keck and space-based images from the
Hubble Space Telescope (HST). The median projected angular (physical) offset is
0.17 arcsec (1.3 kpc). The median offset normalized by the individual host
half-light radii is 0.98 suggesting a strong connection of GRB locations with
the UV light of their hosts. This provides strong observational evidence for
the connection of GRBs to star formation. We compare the observed offset
distribution with an exponential disk, a model for the location of collapsars.
The statistical comparison shows good agreement, with the Kolmogorov-Smirnov
probability that the observed offsets derive from the model distribution of
P_KS = 0.45. We also compare the observed GRB offsets with the expected offset
distribution of delayed merging remnant progenitors (black hole--neutron star
and neutron star--neutron star binaries). We find that delayed merging remnant
progenitors, insofar as the predicted offset distributions from population
synthesis studies are representative, can be ruled out at the 2 x 10^{-3}
level. This is arguably the strongest observational constraint yet against
delayed merging remnants as the progenitors of long-duration GRBs. In the
course of this study, we have also discovered the putative host galaxies of GRB
990510 and GRB 990308 in archival HST data. (abstract abridged)

Astrophysics