  One of the most famous classical tests of General Relativity is the
gravitoelectric secular advance of the pericenter of a test body in the
gravitational field of a central mass. In this paper we explore the possibility
of performing a measurement of the gravitoelectric pericenter advance in the
gravitational field of the Earth by analyzing the laser-ranged data to some
existing, or proposed, laser-ranged geodetic satellites. At the present level
of knowledge of various error sources, the relative precision obtainable with
the data from LAGEOS and LAGEOS II, suitably combined, is of the order of
$10^{\rm -3}$. Nevertheless, these accuracies could sensibly be improved in the
near future when the new data on the terrestrial gravitational field from the
CHAMP and GRACE missions will be available. The use of the perigee of LARES
(LAser RElativity Satellite), in the context of a suitable combination of
orbital residuals including also LAGEOS II, should further raise the precision
of the measurement. As a secondary outcome of the proposed experiment, with the
so obtained value of $\ppn$ and with $\et=4\beta-\gamma-3$ from Lunar Laser
Ranging it could be possible to obtain an estimate of the PPN parameters
$\gamma$ and $\beta$ at the $10^{-2}-10^{-3}$ level.
