  In this paper, we further develop the model for the migration of planets
introduced in Del Popolo et al. (2001). We first model the protoplanetary
nebula as a time-dependent accretion disc and find self-similar solutions to
the equations of the accretion disc that give to us explicit formulas for the
spatial structure and the temporal evolution of the nebula. These equations are
then used to obtain the migration rate of the planet in the planetesimal disc
and to study how the migration rate depends on the disc mass, on its time
evolution and on some values of the dimensionless viscosity parameter alpha. We
find that planets that are embedded in planetesimal discs, having total mass of
10^{-4}-0.1 M_{\odot}, can migrate inward a large distance for low values of
alpha (e.g., alpha \simeq 10^{-3}-10^{-2}) and/or large disc mass and can
survive only if the inner disc is truncated or because of tidal interaction
with the star. Orbits with larger $a$ are obtained for smaller value of the
disc mass and/or for larger values of alpha. This model may explain several
orbital features of the recently discovered giant planets orbiting nearby
stars.
