  In this paper we discuss the evolution of gravitationally unstable
pre-galactic discs that result from the collapse of haloes at high redshift $z
\approx 10$ or so, which have not yet been enriched by metals. In cases where
molecular hydrogen formation is suppressed the discs are maintained at a
temperature of a few thousand degrees Kelvin. However, when molecular hydrogen
is present cooling can proceed down to a few hundred degrees Kelvin. Analogous
to the case of the larger scale proto-galactic discs, we assume that the
evolution of these discs is mainly driven by angular momentum redistribution
induced by the development of gravitational instabilities in the disc. We also
properly take into account the possibility of disc fragmentation. We thus show
that this simple model naturally predicts the formation of supermassive black
holes in the nuclei of such discs and provides a robust determination of their
mass distribution as a function of halo properties. We estimate that roughly 5%
of discs resulting from the collapse of haloes with $M\approx 10^7 M_{\odot}$
should host a massive black hole with a mass $M_{\rm BH}\approx 10^5
M_{\odot}$. We confirm our arguments with time-dependent calculations of the
evolution of the surface density and of the accretion rate in these primordial
discs. This mechanism offers an efficient way to form seed black holes at high
redshift. The predicted masses for our black hole seeds enable the comfortable
assembly of $10^9 M_{\odot}$ black holes powering the luminous quasars detected
by the Sloan Digital Sky Survey at $z = 6$ for a concordance cosmology.
(abridged)
