  Primordial gas in protogalactic dark matter (DM) halos with virial
temperatures Tvir > 10^4 K begins to cool and condense via atomic hydrogen.
Provided this gas is irradiated by a strong ultraviolet (UV) flux and remains
free of H2 and other molecules, it has been proposed that the halo with Tvir
~10^4 K may avoid fragmentation, and lead to the rapid formation of a
supermassive black hole (SMBH) as massive as M=10^5-10^6 Msun. This
``head--start'' would help explain the presence of SMBHs with inferred masses
of several x 10^9 Msun, powering the bright quasars discovered in the Sloan
Digital Sky Survey at redshift z>~6. However, high-redshift DM halos with
Tvir~10^4K are likely already enriched with at least trace amounts of metals
and dust produced by prior star-formation in their progenitors. Here we study
the thermal and chemical evolution of low-metallicity gas exposed to extremely
strong UV radiation fields. Our results, obtained in one-zone models, suggest
that gas fragmentation is inevitable above a critical metallicity, whose value
is between Zcr~3x10^{-4} Zsun (in the absence of dust) and as low as Zcr~ 5 x
10^{-6} Zsun (with a dust-to-gas mass ratio of about 0.01 Z/Zsun). We propose
that when the metallicity exceeds these critical values, dense clusters of
low--mass stars may form at the halo nucleus. Relatively massive stars in such
a cluster can then rapidly coalesce into a single more massive object, which
may produce an intermediate-mass BH remnant with a mass up to M <~10^2-10^3
Msun.
