  We examine the significance of the first metal-free stars (Pop III) for the
cosmological reionization of HI and HeII. These stars have unusually hard
spectra, with the integrated ionizing photon rates from a Pop III stellar
cluster for HI and HeII being 1.6 and $10^5$ times stronger respectively than
those from a Pop II cluster. For the currently favored cosmology, we find that
Pop III stars alone can reionize HI and HeII at redshifts, $z$ of about 9 (4.7)
and 5.1 (0.7) for continuous (instantaneous) modes of star formation. More
realistic scenarios involving combinations of Pop III and Pop II stellar
spectra yield similar results for hydrogen. Helium never reionizes completely
in these cases; the ionization fraction of HeIII reaches a maximum of about 60
percent at $z$ of about 5.6 if Pop III star formation lasts for $10^9$ yr.
Future data on HI reionization can test the amount of small-scale power
available to the formation of the first objects, and provide a constraint on
values of $\sigma_8$ less than or about 0.7. Since current UV observations
indicate an epoch of reionization for HeII at $z$ of about 3, HeII may reionize
more than once. Measurements of the HeII Gunn-Peterson effect in the
intergalactic medium at redshifts exceeding about 3 may reveal the significance
of Pop III stars for HeII reionization, particularly in void regions that may
contain relic ionization from early Pop III stellar activity.
