  We calculate in this work the rates for the neutrino pair production by
nucleon-nucleon bremsstrahlung taking into account the full contribution from a
nuclear one-pion-exchange potential. It is shown that if the temperatures are
low enough ($T \leq 20 MeV$), the integration over the nuclear part can be done
for the general case, ranging from the completely degenerate (D) to the
non-degenerate (ND) regime. We find that the inclusion of the full nuclear
contribution enhances the neutrino pair production by $nn$ and $pp$
bremsstrahlung by a factor of about two in both the D and ND limits when
compared with previous calculations. This result may be relevant for the
physical conditions of interest in the semitransparent regions near the
neutrinosphere in type II supernovae, cooling of neutron stars and other
astrophysical situations.
