  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 fates of dust grains within SNRs heavily
depend on their initial radii $a_{\rm ini}$. For Type II SNRs expanding into
the ambient medium with density of $n_{\rm H,0} = 1$ cm$^{-3}$, grains of
$a_{\rm ini} < 0.05$ $\mu$m are detained in the shocked hot gas and are
completely destroyed, while grains of $a_{\rm ini} > 0.2$ $\mu$m are injected
into the surrounding medium without being destroyed significantly. Grains with
$a_{\rm ini}$ = 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$^{-6}-10^{-4}$ $Z_\odot$, high enough to form low-mass stars
with 0.1-1 $M_\odot$. 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
impacts on the stellar mass and elemental composition of Population II.5 stars
formed in the shell of the SNR.
