We propose that the cosmological gamma-ray bursts arise from the collapse of neutron stars to black holes, which is triggered by collisions or mergers with less evolved stars. This scenario represents a cosmological history qualitatively different from most previous theories because it contains a significant contribution from an old stellar population, namely the globular clusters. Furthermore, the gas-poor central regions of globular clusters provide an ideal environment for the generation of the recently confirmed afterglows via the fireball scenario. Collisions in close binaries resulting from neutron star birth kicks may also contribute to the overall rate and should lead to associations between some gamma-ray bursts and supernovae of Type Ib/c.