  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, in a more consistent way than in earlier
publications. It is shown in part 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 upper bound
on the neutrino magnetic moment from SN1987A is improved by the factor of 3 to
7.
