We investigate the evolution of dust formed in Population III supernovae (SNe) by considering its transport and processing by sputtering within the SN remnants (SNRs). We find that the fate of dust grains within SNRs heavily depends on their initial radii aini. For Type II SNRs expanding into the ambient medium with density of nH,0 = 1 cm−3, grains of aini < 0.05 μm are detained in the shocked hot gas and are completely destroyed, while grains of aini > 0.2 μm are injected into the surrounding medium without being significantly destroyed. Grains with aini = 0.05–0.2 μm are finally trapped in the dense shell behind the forward shock. We show that the grains piled up in the dense shell enrich the gas up to 10−6–10−4Z⊙, high enough to form low-mass stars with 0.1–1 M⊙. In addition, [Fe/H] in the dense shell ranges from −6 to −4.5, which is in good agreement with the ultra-metal-poor stars with [Fe/H] < −4. We suggest that newly formed dust in a Population III SN can have great impact on the stellar mass and elemental composition of Population II.5 stars formed in the shell of the SNR.