We examine the significance of the first metal-free stars (Population Ⅲ) for the cosmological reionization of H Ⅰ and He Ⅱ. These stars have unusually hard spectra, with the integrated ionizing photon rates from a Population Ⅲ stellar cluster for H Ⅰ and He Ⅱ being 1.6 and 10~5 times stronger, respectively, than those from a Population Ⅱ cluster. For the currently favored cosmology, we find that Population Ⅲ stars alone can reionize H Ⅰ at redshifts of z approx= 9 and 4.7 and He n at z approx= 5.1 and 0.7 for continuous and instantaneous modes of star formation, respectively. More realistic scenarios involving combinations of Population Ⅲ and Population Ⅱ stellar spectra yield similar results for hydrogen. Helium never reionizes completely in these cases; the ionization fraction of He in reaches a maximum of about 60% at z ～ 5.6 if Population Ⅲ star formation lasts for 10~9 yr. Future data on H Ⅰ reionization can test the amount of small-scale power available to the formation of the first objects and provide a constraint on values of σ_8 approx< 0.7. Since current UV observations indicate an epoch of reionization for He Ⅱ at z ～ 3, He Ⅱ may reionize more than once. Measurements of the He Ⅱ Gunn-Peterson effect in the intergalactic medium at redshifts z approx> 3 may reveal the significance of Population Ⅲ stars for He Ⅱ reionization, particularly in "void" regions that may contain "relic" ionization from early Population Ⅲ stellar activity.
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