The evolution of neutron stars in close binary systems with a low-mass
companion is considered assuming the magnetic field to be confined within the
solid crust. We adopt the standard scenario of the evolution in a close binary
system in accordance with which the neutron star passes throughout four
evolutionary phases ("isolated pulsar" -- "propeller" -- accretion from the
wind of a companion -- accretion due to Roche-lobe overflow). Calculations have
been performed for a great variety of parameters characterizing the properties
both of the neutron star and low-mass companion. We find that neutron stars
with more or less standard magnetic field and spin period being processed in
low-mass binaries can evolve to low-field rapidly rotating pulsars. Even if the
main-sequence life of a companion is as long as 1010 yr, the neutron star
can maintain a relatively strong magnetic field to the end of the accretion
phase. The considered model can well account for the origin of millisecond
pulsars.Comment: 18 pages + 10 figures, uses epsf.sty. Accepted by MNRA