  We investigate the condition for the formation of low-mass second-generation
stars in the early universe. It has been proposed that gas cooling by dust
thermal emission can trigger fragmentation of a low-metallicity star-forming
gas cloud. In order to determine the critical condition in which dust cooling
induces the formation of low-mass stars, we follow the thermal evolution of a
collapsing cloud by a one-zone semi-analytic collapse model. Earlier studies
assume the dust amount in the local universe, where all refractory elements are
depleted onto grains, and/or assume the constant dust amount during gas
collapse. In this paper, we employ the models of dust formation and destruction
in early supernovae to derive the realistic dust compositions and size
distributions for multiple species as the initial conditions of our collapse
calculations. We also follow accretion of heavy elements in the gas phase onto
dust grains, i.e., grain growth, during gas contraction. We find that grain
growth well alters the fragmentation property of the clouds, and that this
still does not approach to the value in the local universe. The critical
conditions can be written by the gas metallicity Zcr and the initial depletion
efficiency fdep,0 of gas-phase metal onto grains, or dust-to-metal mass ratio,
as (Zcr/10^{-5.5} Zsun) = (fdep,0/0.18)^{-0.44} with small scatters in the
range of Zcr = [0.06--3.2]x10^{-5} Zsun. We also show that the initial dust
composition and size distribution are important to determine Zcr.
