The evolution of a gas shell, swept up by the supernova remnant of a massive first-generation star, is studied with H2 and HD chemistry taken into account and with the use of a semianalytical approximation to the dynamics. When a first-generation star, formed in a parent pregalactic cloud, explodes as a supernova with explosion energy in the range of 1051-1052 ergs at redshifts of z = 10-50, H2 and HD molecules are formed in the swept up gas shell at fractional abundances of ~10-3 and ~10-5, respectively, and effectively cool the gas shell to temperatures of 32-154 K. If the supernova remnant can sweep to gather the ambient gas of mass 6 × 104 to 8 × 105 M☉, the gas shell comes to be dominated by its self-gravity and, hence, is expected to fragment. The amount of swept up gas necessary for fragmentation increases with the explosion energy and decreases with the interstellar gas density (or redshift) of the host cloud, which provides a lower boundary to the mass of the host cloud in which star formation is triggered by the first-generation supernova. Also, the condition for fragmentation is very sensitive to the thermal state of interstellar gas. Our result shows that for a reasonable range of temperatures (200-1000 K) of interstellar gas, the formation of second-generation stars can be triggered by a single supernova or hypernova with explosion energy in the above range in a primordial cloud of total (dark and baryonic) mass as low as a few times 106 M☉. For higher temperatures in the interstellar gas, however, the condition for the fragmentation in the swept up gas shell demands a larger supernova explosion energy. We also follow the subsequent contraction of the fragment pieces assuming their geometry (sphere and cylinder) and demonstrate that the Jeans masses in the fragments decrease to well below 1 M☉ by the time the fragments become optically thick to the H2 and HD lines. The fragments are then expected to break up into dense cores whose masses are comparable to the Jeans masses and collapse to form low-mass stars that can survive to the present. If the material in the gas shell is mixed well with the ejecta of the supernova, the shell and low-mass stars thus formed are likely to have metals of abundance [Fe/H] -3 on average. This metallicity is consistent with those of the extremely metal-poor stars found in the Galactic halo. Stars with low metallicities of [Fe/H] -5 such as HE 0107-5240, recently discovered in the Galactic halo, are difficult to form by this mechanism and must be produced in different situations.
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