We differentiate between the metal enrichment of the gas in virialized minihalos and that of the intergalactlc medium at high redshift, which is pertinent to cosmological reionization, with the initial expectation that gas in the high-density regions within formed dark matter halos may be more robust and thus resistant to mixing with the lower density intergalactic medium. Using detailed hydrodynamic simulations of gas clouds in minihalos subject to destructive processes associated with the encompassing intergalactic shocks carrying metal-enriched gas, we find, as an example, that, for realistic shocks with velocities of 10-100 km s super(-1), more than (90%, 65%) of the high-density gas with rho greater than or equal to 500 rho b inside a minihalo virialized at z = 10 with a mass of (10 super(7), 10 super(6)) M unk, respectively, remains at a metalliclty lower than 3% of that of the intergalactic medium by redshift z = 6. It may be expected that the high-density gas in minihalos will become fuel for subsequent star formation when they are incorporated into larger halos, where efficient atomic cooling can induce gas condensation and hence star formation. Since minihalos vlrialize at high redshift, when the universe is not expected to have been significantly reionized, the implication is that gas in virialized minihalos may provide an abundant reservoir of primordial gas that could possibly allow the formation of Population III metal-free stars to extend to much lower redshifts than would have been otherwise expected on the basis of the enrichment of the intergalactic medium.