  We consider the amplification and transport of a magnetic field in the
collapsed core of a massive star, including both the region between the
neutrinosphere and the shock, and the central, opaque core. An analytical
argument explains why rapid convective overturns persist within a newly formed
neutron star for roughly 10 seconds ($> 10^3$ overturns), consistent with
recent numerical models. A dynamical balance between turbulent and magnetic
stresses within this convective layer corresponds to flux densities in excess
of $10^{15}$G. Material accreting onto the core is heated by neutrinos and also
becomes strongly convective. We compare the expected magnetic stresses in this
convective `gain layer' with those deep inside the neutron core.
  Buoyant motions of magnetized fluid are greatly aided by the intense neutrino
flux. We calculate the transport rate through a medium containing free neutrons
protons, and electrons, in the limiting cases of degenerate or non-degenerate
nucleons. Fields stronger than $\sim 10^{13}$ G are able to rise through the
outer degenerate layers of the neutron core during the last stages of
Kelvin-Helmholtz cooling (up to 10 seconds post-collapse), even though these
layers have become stable to convection. We also find the equilibrium shape of
a thin magnetic flux rope in the dense hydrostatic atmosphere of the neutron
star, along with the critical separation of the footpoints above which the rope
undergoes unlimited expansion against gravity. The implications of these
results for pulsar magnetism are summarized, and applied to the case of late
fallback over the first 1,000-10,000 s of the life of a neutron star
