  A linear nonadiabatic analysis of the vibrational stability of population III
main-sequence stars was carried out. It was demonstrated that, in the case of
massive stars with $M \gtrsim 5\Mo$, helium burning (triple alpha reaction)
starts during the main-sequence stage and produces $^{12}{\rm C}$, leading to
the activation of a part of the CNO-cycle. It was found that, despite of that,
those stars with $M\lesssim 13\Mo$ become unstable against the dipole g$_1$-
and g$_2$-modes due to the $\varepsilon$-mechanism, during the early
evolutionary phase at which the pp-chain is still the dominant nuclear energy
source. The instability due to the $\varepsilon$-mechanism occurs against
g-modes having a large amplitude in the off-centered $^3$He accumulation shell
in the deep interior, and the growth time is much shorter than the evolutionary
timescale. This instability is therefore likely to induce mixing in the stellar
interior, and have a significant influence on the evolution of these stars.
