Carbon-enhanced metal-poor ( CEMP ) stars are the living fossils holding records 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 gravitational collapse of the enriched cloud , considering all relevant cooling/heating processes and chemical reactions as well as the growth of dust grains . We adopt faint SN models for the first time with progenitor masses M _ { PopIII } = 13 – 80 { M _ { \bigodot } } , which yield C-enhanced abundance patterns ( { [ { C } / Fe ] } = 4.57 – 4.75 ) through mixing and fallback of innermost layers of the ejecta . This model also considers the formation and destruction of dust grains . We find that the metals ejected by the SN can be partly re-accreted by the same dark matter minihalo , and carbon abundance of the enriched cloud A ( { C } ) = 3.80 – 5.06 is lower than the abundance range of observed CEMP stars ( A ( { C } ) \gtrsim 6 ) because the mass of the metals ejected by faint SNe is smaller than normal core-collapse SNe due to extensive fallback . We also find that cloud fragmentation is induced by gas cooling from carbonaceous grains for M _ { PopIII } = 13 { M _ { \bigodot } } even with the lowest iron abundance { [ { Fe } / H ] } \sim - 9 . This leads to the formation of low-mass stars , and these “ giga metal-poor ” stars can survive until the present-day Universe and may be found by future observations .