  In the early universe, Dirac neutrino magnetic moments due to their
chirality-flipping nature could lead to thermal production of right-handed
neutrinos, which would make a significant contribution to the effective
neutrino number, $N_{\rm eff}$. We present in this paper a dedicated
computation of the neutrino chirality-flipping rate in the thermal plasma. With
a careful and consistent treatment of soft scattering and the plasmon effect in
finite temperature field theories, we find that neutrino magnetic moments above
$2.7\times 10^{-12}\mu_B$ have been excluded by current CMB and BBN
measurements of $N_{\rm eff}$, assuming flavor-universal and diagonal magnetic
moments for all three generation of neutrinos. This limit is stronger than the
latest bounds from XENONnT and LUX-ZEPLIN experiments, and comparable with
those from stellar cooling considerations.
