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 similar or equal to 9 and 4.7 and He II at z similar or equal to 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 III reaches a maximum of about 60% at z similar to 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 sigma(8) less than or similar to 0.7. Since current UV observations indicate an epoch of reionization for He II at z similar to 3, He II may reionize more than once. Measurements of the He II Gunn-Peterson effect in the intergalactic medium at redshifts z greater than or similar to 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.