We report evidence from the Third BATSE Gamma-Ray Burst Catalog that long (T90> 10 s) and short (T90 < 10 s) gamma-ray bursts (GRBs) represent distinct source populations. Their spatial distributions are significantly different, with long bursts having <V/Vmax> = 0.282±0.014 but short bursts having <V/Vmax> = 0.385±0.019, differing by 0.103±0.024, significant at the 4.3 σ level. This implies different spatial origin and physical processes for long and short bursts. Long bursts may be explained by accretion-induced collapse. Short bursts require another mechanism, for which we suggest neutron star collisions. These are capable of producing neutrino bursts as short as a few milliseconds, consistent with the shortest observed timescales in GRBs. We briefly investigate the parameters of clusters in which neutron star collisions may occur, and discuss the nuclear evolution of expelled and accelerated matter.