  Spectroscopic and photometric observations show that many globular clusters
host multiple stellar populations, challenging the common paradigm that
globular clusters are "simple stellar populations" composed of stars of uniform
age and chemical composition. The chemical abundances of second-generation (SG)
stars constrain the sources of gas out of which these stars must have formed,
indicating that the gas must contain matter processed through the
high-temperature CNO cycle. First-generation massive Asymptotic Giant Branch
(AGB) stars have been proposed as the source of this gas. In a previous study,
by means of hydrodynamical and N-body simulations, we have shown that the AGB
ejecta collect in a cooling flow in the cluster core, where the gas reaches
high densities, ultimately forming a centrally concentrated subsystem of SG
stars. In this Letter we show that the high gas density can also lead to
significant accretion onto a pre-existing seed black hole. We show that gas
accretion can increase the black hole mass by up to a factor of 100. The
details of the gas dynamics are important in determining the actual black hole
growth. Assuming a near-universal seed black hole mass and small
cluster-to-cluster variations in the duration of the SG formation phase, the
outcome of our scenario is one in which the present intermediate-mass black
hole (IMBH) mass may have only a weak dependence on the current cluster
properties. The scenario presented provides a natural mechanism for the
formation of an IMBH at the cluster center during the SG star-formation phase.
