Abstract The hypercritical accretion described by Chevalier in the preceding article forces a re-examination of the evolution of binary pulsars. We favor helium star binaries as progenitors, so that the first neutron star does not have to go through a hydrogen envelope. In this way it avoids going into a black hole. We argue here that, if the neutron-star helium-star binary following the first explosion is narrow, with relative axis a ~ 1 R ⊙, then accretion from the helium star can bring the magnetic field of the neutron star down from B ≥ 10 12 G to B ~ 10 10 G. The neutron-star neutron-star binary resulting from such a narrow progenitor is likely to also be relatively narrow, although there is a tendency in the explosions of the remaining helium star for the orbit to open up by a factor of ~ (1 − e) −1, where e is the eccentricity of the relativistic neutron-star binary. Something like PSR 1913 + 16 or PSR 1534 + 12 results. Because the magnetic fields of the pulsars have been so reduced by accretion, their spin-down times are greatly increased, and, therefore, also the probability of seeing them.