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Carbon-enhanced metal-poor (CEMP) stars are the living fossils holdingrecords of chemical enrichment from early generations of stars. In this work,we perform a set of numerical simulations of the enrichment from a supernova(SN) of a first generation of metal-free (Pop III) star and the gravitationalcollapse of the enriched cloud, considering all relevant cooling/heatingprocesses and chemical reactions as well as the growth of dust grains. We adoptfaint SN models for the first time with progenitor masses $M_{\rm PopIII} =13$--$80 \ {\rm M}_{\bigodot}$, which yield C-enhanced abundance patterns(${\rm [C/Fe]} = 4.57$--$4.75$) through mixing and fallback of innermost layersof the ejecta. This model also considers the formation and destruction of dustgrains. We find that the metals ejected by the SN can be partly re-accreted bythe same dark matter minihalo, and carbon abundance of the enriched cloud$A({\rm C}) = 3.80$--$5.06$ is lower than the abundance range of observed CEMPstars ($A({\rm C}) \gtrsim 6$) because the mass of the metals ejected by faintSNe is smaller than normal core-collapse SNe due to extensive fallback. We alsofind that cloud fragmentation is induced by gas cooling from carbonaceousgrains for $M_{\rm PopIII} = 13 \ {\rm M}_{\bigodot}$ even with the lowest ironabundance ${\rm [Fe/H]} \sim -9$. This leads to the formation of low-massstars, and these ``giga metal-poor'' stars can survive until the present-dayUniverse and may be found by future observations.

Seeding the second star -- II. CEMP star formation enriched from faint supernovae