  We consider the expulsion of the magnetic field from the super-conducting
core of a neutron star and its subsequent decay in the crust. Particular
attention is paid to a strong feedback of the distortion of magnetic field
lines in the crust on the expulsion of the flux from the core. This causes a
considerable delay of the core flux expulsion if the initial field strength is
larger than 10^{11} G. It is shown that the hypothesis on the magnetic field
expulsion induced by the neutron star spin-down is adequate only for a
relatively weak initial magnetic field $B \approx 10^{11}$ G. The expulsion
time-scale depends not only on the conductivity of the crust, but also on the
initial magnetic field strength itself. Our model of the field evolution
naturally explains the existence of the residual magnetic field of neutron
stars. Its strength is correlated with the impurity concentration in neutron
star crusts and anti-correlated with the initial field strengths.
