  We investigate the origin of carbon-enhanced metal-poor (CEMP) stars starting
from the recently discovered $\rm [Fe/H]<-7.1$ star SMSS J031300 (Keller et al.
2014). We show that the elemental abundances observed on the surface of SMSS
J031300 can be well fit by the yields of faint, metal free, supernovae. Using
properly calibrated faint supernova explosion models, we study, for the first
time, the formation of dust grains in such carbon-rich, iron-poor supernova
ejecta. Calculations are performed assuming both unmixed and uniformly mixed
ejecta and taking into account the partial destruction by the supernova reverse
shock. We find that, due to the paucity of refractory elements beside carbon,
amorphous carbon is the only grain species to form, with carbon condensation
efficiencies that range between (0.15-0.84), resulting in dust yields in the
range (0.025-2.25)M$_{\odot}$. We follow the collapse and fragmentation of a
star forming cloud enriched by the products of these faint supernova explosions
and we explore the role played by fine structure line cooling and dust cooling.
We show that even if grain growth during the collapse has a minor effect of the
dust-to-gas ratio, due to C depletion into CO molecules at an early stage of
the collapse, the formation of CEMP low-mass stars, such as SMSS J031300, could
be triggered by dust cooling and fragmentation. A comparison between model
predictions and observations of a sample of C-normal and C-rich metal-poor
stars supports the idea that a single common pathway may be responsible for the
formation of the first low-mass stars.
