We investigate the evolution of dust formed in Population III supernovae (SNe) by considering its transport mid processing by sputtering within the SN remnants (SNRs). We find that; the fate of dust grains within SNRs heavily depends on their initial radii a(ini). For Type II SNRs expanding into the ambient medium with density of n(H,0) = 1 cm(-3), grains of a(ini) < 0.05 mu m SNRs expanding into the ambient medium with density of n(H,0) = 1 cm(-3) are detained in the shocked hot gas and are completely destroyed, while grains of a(ini) > 0.2 mu m are injected into the surrounding medium without being significantly destroyed. Grains with a(ini) =0.05-0.2 mu m 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(-1) Z(circle dot), high enough to form low-mass, stars with 0.1-1 M-circle dot. 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.