  In proto-neutron stars with strong magnetic fields, the neutrino-nucleon
scattering/absorption cross sections depend on the direction of neutrino
momentum with respect to the magnetic field axis, a manifestation of parity
violation in weak interactions. We study the deleptonization and thermal
cooling (via neutrino emission) of proto-neutron stars in the presence of such
asymmetric neutrino opacities. Significant asymmetry in neutrino emission is
obtained due to multiple neutrino-nucleon scatterings. For an ordered magnetic
field threading the neutron star interior, the fractional asymmetry in neutrino
emission is about $0.006 (B/10^{14}G)$, corresponding to a pulsar kick velocity
of about $200 (B/10^{14}G)$ km/s for a total radiated neutrino energy of
$3\times 10^{53}$ erg.
