  We consider the equilibration of the 'wrong--helicity' Dirac neutrino states
$\nu_{+}$ and $\overline\nu_{-}$ in the early Universe via weak interactions
and calculate carefully the thermally averaged production rate, taking into
account all the relevant scattering and decay processes. Requiring that the
production rate is less than the Hubble parameter at the onset of QCD phase
transition so that the nucleosynthesis predictions are not contradicted, we
find for $T_{\rm{QCD}}\simeq 200$ MeV the upper limits $m_{\nu_\tau}\lsim 740$
keV and $m_{\nu_\mu}\lsim 480$ keV.
