  We utilize cosmological hydrodynamic simulations to study the formation of
Population III (Pop III) stars in dark matter halos exposed to strong ionizing
radiation. We simulate the formation of three halos subjected to a wide range
of ionizing fluxes, and find that for high flux, ionization and photoheating
can delay gas collapse and star formation up to halo masses significantly
larger than the atomic cooling threshold. The threshold halo mass at which gas
first collapses and cools increases with ionizing flux for intermediate values,
and saturates at a value approximately an order of magnitude above the atomic
cooling threshold for extremely high flux (e.g. $\approx 5 \times 10^8 ~
M_\odot$ at $z\approx6$). This behavior can be understood in terms of
photoheating, ionization/recombination, and Ly$\alpha$ cooling in the
pressure-supported, self-shielded gas core at the center of the growing dark
matter halo. We examine the spherically-averaged radial velocity profiles of
collapsing gas and find that a gas mass of up to $\approx 10^{6}~ M_\odot$ can
reach the central regions within $3~{\rm Myr}$, providing an upper limit on the
amount of massive Pop III stars that can form. The ionizing radiation increases
this limit by a factor of a few compared to strong Lyman-Werner (LW) radiation
alone. We conclude that the bright HeII 1640 \AA\ emission recently observed
from the high-redshift galaxy CR7 cannot be explained by Pop III stars alone.
However, in some halos, a sufficient number of Pop III stars may form to be
detectable with future telescopes such as the James Webb Space Telescope
(JWST).
