In this paper, we discuss the evolution of gravitationally unstable pre-galactic discs that result from the collapse of haloes at high redshift z≈ 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 Kelvin. However, when molecular hydrogen is present, cooling can proceed down to a few hundred Kelvin. Analogous to the case of the larger-scale protogalactic 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 per cent of discs resulting from the collapse of haloes with M≈ 107 M⊙ should host a massive black hole with a mass MBH≈ 105 M⊙. We confirm our arguments with time-dependent calculations of the evolution of the surface density and of the accretion rate in these primordial discs. The luminosity of the outer, colder disc is expected to be very low (in the range of a few thousand L⊙), while the formation of the black hole is expected to produce a burst with a luminosity of a few times 109 L⊙. 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 109-M⊙ black holes powering the luminous quasars detected by the Sloan Digital Sky Survey at z= 6 for a concordance cosmology.