We present a three-dimensional numerical simulation that resolves the formation process of a Population III star down to a scale of ~100 AU. The simulation is initialized on the scale of a dark matter halo of mass ~10^6 M_sun that virializes at z~20. It then follows the formation of a fully-molecular central core, and traces the accretion from the diffuse dust-free cloud onto the protostellar core for as long as ~10^4 yr, at which time the core has grown to ~50 M_sun. We find that the accretion rate starts very high, ~0.1 M_sun yr^-1, and declines rapidly thereafter approaching a power-law temporal scaling. Asymptotically, at times t > 10^3 yr after core formation, the stellar mass grows approximately as M_star=20 M_sun (t/10^3 yr)^0.4. Earlier on, accretion is faster with M_star \propto t^0.75. By extrapolating this growth over the full lifetime of very massive stars, t~3x10^6 yr, we obtain the conservative upper limit M_star < 500 M_sun. The actual stellar mass is, however, likely to be significantly smaller than this mass limit due to radiative and mechanical feedback from the protostar.

New Astronomy, in press, June 2004, vol.9, pp.353-364