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 has a mass less than 0.8 M⊙ and a metallicity of Z= 4.5 × 10−5 Z⊙. 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 (SNe) with metal-free progenitors of 20 and 35 M⊙. Using these selected SN explosion models, we compute the corresponding dust yields and the resulting dust depletion factor taking into account the partial destruction by the SN 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 M⊙ 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 M⊙ of dust condense in the first SNe, allowing for moderate destruction by the reverse shock. If dust formation in the first SNe 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 M⊙ 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 SNe and played a fundamental role in the formation of the first low-mass stars.