  We use cosmological hydrodynamic simulations with unprecedented resolution to
study the formation of primordial stars in an ionized gas at high redshifts.
Our approach includes all the relevant atomic and molecular physics to follow
the thermal evolution of a prestellar gas cloud to very high densities of
~10^{18} cm^{-3}. We locate a star-forming gas cloud within a reionized region
in our cosmological simulation. The first run-away collapse is triggered when
the gas cloud's mass is ~40 Msun. We show that the cloud core remains stable
against chemo-thermal instability and also against gravitational deformation
throughout its evolution. Consequently, a single proto-stellar seed is formed,
which accretes the surrounding hot gas at the rate ~10^{-3} Msun/year. We carry
out proto-stellar evolution calculations using the inferred accretion rate. The
resulting mass of the star when it reaches the zero-age main sequence is M_ZAMS
~40 Msun. We argue that, since the obtained M_ZAMS is as large as the mass of
the collapsing parent cloud, the final stellar mass should be close to this
value. Such massive, rather than exceptionally massive, primordial stars are
expected to cause early chemical enrichment of the Universe by exploding as
black hole-forming super/hypernovae, and may also be progenitors of high
redshift gamma-ray bursts. The elemental abundance patterns of recently
discovered hyper metal-poor stars suggest that they might have been born from
the interstellar medium that was metal-enriched by supernovae of these massive
primordial stars.
