  The relative importance of metals and dust grains in the formation of the
first low-mass stars has been a subject of debate. The recently discovered
Galactic halo star SDSS J102915+172927 (Caffau et al. 2011) has a mass less
than 0.8 Msun and a metallicity of Z = 4.5 10^{-5} Zsun. We investigate the
origin and properties of this star by reconstructing the physical conditions in
its birth cloud. We show that the observed elemental abundance trend of SDSS
J102915+172927 can be well fitted by the yields of core-collapse supernovae
with metal-free progenitors of 20 Msun and 35 Msun. Using these selected
supernova explosion models, we compute the corresponding dust yields and the
resulting dust depletion factor taking into account the partial destruction by
the supernova reverse shock. We then follow the collapse and fragmentation of a
star forming cloud enriched by the products of these SN explosions at the
observed metallicity of SDSS J102915+172927. We find that [0.05 - 0.1] Msun
mass fragments, which then lead to the formation of low-mass stars, can occur
provided that the mass fraction of dust grains in the birth cloud exceeds 0.01
of the total mass of metals and dust. This, in turn, requires that at least 0.4
Msun of dust condense in the first supernovae, allowing for moderate
destruction by the reverse shock. If dust formation in the first supernovae is
less efficient or strong dust destruction does occur, the thermal evolution of
the SDSS J102915+172927 birth cloud is dominated by molecular cooling, and only
> 8 Msun fragments can form. We conclude that the observed properties of SDSS
J102915+172927 support the suggestion that dust must have condensed in the
ejecta of the first supernovae and played a fundamental role in the formation
of the first low-mass stars.
