  We describe results from a fully self-consistent three dimensional
hydrodynamical simulation of the formation of one of the first stars in the
Universe. Dark matter dominated pre-galactic objects form because of
gravitational instability from small initidal density perturbations. As they
assemble via hierarchical merging, primordial gas cools through ro-vibrational
lines of hydrogen molecules and sinks to the center of the dark matter
potential well. The high redshift analog of a molecular cloud is formed. When
the dense, central parts of the cold gas cloud become self-gravitating, a dense
core of approximately 100 solar mass undergoes rapid contraction. At densities
n>10^9 cm^-3 a one solar mass proto-stellar core becomes fully molecular due to
three-body H_2 formation. Contrary to analytical expectations this process does
not lead to renewed fragmentation and only one star is formed. The calculation
is stopped when optical depth effects become important, leaving the final mass
of the fully formed star somewhat uncertain. At this stage the protostar is
acreting material very rapidly (~0.01 solar masses per year). Radiative
feedback from the star will not only halt its growth but also inhibit the
formation of other stars in the same pre-galactic object (at least until the
first star ends its life, presumably as a supernova). We conclude that at most
one massive (M >> 1 solar mass) metal free star forms per pre-galactic halo,
consistent with recent abundance measurements of metal poor galactic halo
stars.
