  The neutrino chirality-flip process under the conditions of the supernova
core is investigated in detail with the plasma polarization effects in the
photon propagator taken into account. It is shown that the contribution of the
proton fraction of plasma is essential. New upper bounds on the neutrino
magnetic moment are obtained: mu_nu < (0.5 - 1.1) 10^{-12} mu_B from the limit
on the supernova core luminosity for nu_R emission, and mu_nu < (0.4 - 0.6)
10^{-12} mu_B from the limit on the averaged time of the neutrino spin-flip.
The best astrophysical upper bound on the neutrino magnetic moment is improved
by the factor of 3 to 7.
