  We note that current observational evidence strongly favors a conventional
recombination of ionized matter subsequent to redshift z=1200, followed by
reionization prior to redshift z=5 and compute how this would have occurred in
a standard scenario for the growth of structure. Extending prior semi-analytic
work, we show by direct, high-resolution numerical simulations (of a COBE
normalized CDM+Lambda model) that reheating, will occur in the interval 15>z>7,
followed by reionization and accompanied by a significant increase in the Jeans
mass. However, the evolution of the Jeans mass does not significantly affect
star formation in dense, self-shielded clumps of gas, which are detached from
the thermal evolution of the rest of the universe. On average, the growth of
the Jeans mass tracks the growth of the nonlinear mass scale, a result we
suspect is due to nonlinear feedback effects. Cooling on molecular hydrogen
leads to a burst of star formation prior to reheating which produces Population
III stars with Omega_* reaching 10^{-5.5} and Z/Z_sun reaching 10^{-3.7} by
z=14. Star formation subsequently slows down as molecular hydrogen is depleted
by photo-destruction and the rise of the temperature. At later times, z<10,
when the characteristic virial temperature of gas clumps reach 10,000 degrees,
star formation increases again as hydrogen line cooling become efficient.
Objects containing Pop III stars accrete mass with time and, as soon as they
reach 10,000 K virial temperature, they engage in renewed star formation and
turn into normal Pop II objects having an old Pop III metal poor component.
