  We show that the explosion of the first supernovae can trigger low-mass star
formation via gravitational fragmentation of the supernova-driven gas shell. If
the shell mass does not exceed the host galaxy gas mass, all explosions with
energies E_SN > 10^{51} erg can lead to shell fragmentation. However, the
minimum ambient density required to induce such fragmentation is much larger,
n_0 > 300 cm^{-3}, for Type II supernovae than for pair-instability ones, which
can induce star formation even at lower ambient densities. The typical mass of
the unstable fragments is 10^{4-7} Msun; their density is in the range
110-6x10^7 cm{-3}. Fragments have a metallicity strictly lower than 10^{-2.6}
Zsun and large values of the gravitational-to-pressure force ratio f ~ 8. Based
on these findings, we conclude that the second generation of stars produced by
such self-propagating star formation is predominantly constituted by low-mass,
long-living, extremely metal-poor (or even metal-free, if mixing is suppressed)
stars. We discuss the implications of such results for Pop III star formation
scenarios and for the most iron-poor halo star HE0107-5240.
