FOLLOWING the discovery of X-ray sources in globular clusters, the accretion of matter onto a central massive black hole was suggested1-3 as a possible explanation. Subsequently, it was found4,5 that these sources could be readily explained by thermonuclear instabilities on neutron-star surfaces and the black-hole models were abandoned. We show here, however, that the recent discovery6 of large populations of millisecond pulsars-and hence neutron stars-in globular clusters implies that several hundred stellar black holes (of about ten solar masses) should form within a typical cluster. In clusters of high central density, we find that the rapid dynamical evolution of the black-hole population will cause ejection of nearly all of the holes on a relatively short time-scale. But in systems of intermediate density, some of the surviving holes may capture a normal star to form a low-mass X-ray binary. We suggest that there may be one or more such binaries in the globular clusters surrounding our Galaxy. These systems will be quiescent most of the time-with only occasional X-ray outbursts-but future observations of the hard X-ray spectrum may indirectly establish their existence.