  Dust grains in low-metallicity star-forming regions may be responsible for
the formation of the first low-mass stars. The minimal conditions to activate
dust-induced fragmentation require the gas to be pre-enriched above a critical
dust-to-gas mass ratio Dcr=[2.6--6.3]x10^-9 with the spread reflecting the
dependence on the grain properties. The recently discovered Galactic halo star
SDSS J102915+172927 has a stellar mass of 0.8 Msun and a metallicity of
Z=4.5x10^-5 Zsun and represents an optimal candidate for the dust-induced
low-mass star formation. Indeed, for the two most plausible Population III
supernova progenitors, with 20 Msun and 35 Msun, the critical dust-to-gas mass
ratio can be overcome provided that at least 0.4 Msun of dust condenses in the
ejecta, allowing for moderate destruction by the reverse shock. Here we show
that even if dust formation in the first supernovae is less efficient or strong
dust destruction does occur, grain growth during the collapse of the parent gas
cloud is sufficiently rapid to activate dust cooling and likely fragmentation
into low-mass and long-lived stars. Silicates and magnetite grains can
experience significant grain growth in the density range 10^9 /cc < nH<10^12
/cc by accreting gas-phase species (SiO, SiO2, and Fe) until their gas-phase
abundance drops to zero, reaching condensation efficiencies =1. The
corresponding increase in the dust-to-gas mass ratio allows dust-induced
cooling and fragmentation to be activated at 10^12 /cc < nH < 10^14 /cc, before
the collapsing cloud becomes optically thick to continuum radiation. We show
that for all the initial conditions that apply to the parent cloud of SDSS
J102915+172927, dust-driven fragmentation is able to account for the formation
of the star.
