  We have investigated the turbulent mean-field dynamo action in protoneutron
stars that are subject to convective and neutron finger instabilities during
the early evolutionary phase. While the first one develops mostly in the inner
regions of the star, the second one is favored in the outer regions, where the
Rossby number is much smaller and a mean-field dynamo action is more efficient.
By solving the mean-field induction equation we have computed the critical spin
period below which no dynamo action is possible and found it to be $\sim 1$ s
for a wide range of stellar models and for both axisymmetric and
non-axisymmetric magnetic fields. Because this critical period is substantially
longer than the characteristic spin period of very young pulsars, we expect
that a mean-field dynamo will be effective for most protoneutron stars. The
saturation dipole field estimated by making use of the model of ``global''
quenching fits well the pulsar magnetic fields inferred from the spin-down
data. Apart from the large scale magnetic field, our model predicts also a
generation of small scale fields which are typically stronger than the poloidal
field and can survive during the lifetime of pulsars. Extremely rapidly
rotating protoneutron stars ($P \sim 1$ ms) may have the dipole field $\sim
(3-6) \times 10^{14}$ G.
