  In this paper the effect of the post-Newtonian gravitomagnetic force on the
mean longitudes $l$ of a pair of counter-rotating Earth artificial satellites
following almost identical circular equatorial orbits is investigated. The
possibility of measuring it is examined. The observable is the difference of
the times required to $l$ in passing from 0 to 2$\pi$ for both senses of
motion. Such gravitomagnetic time shift, which is independent of the orbital
parameters of the satellites, amounts to 5$\times 10^{-7}$ s for Earth; it is
cumulative and should be measured after a sufficiently high number of
revolutions. The major limiting factors are the unavoidable imperfect
cancellation of the Keplerian periods, which yields a constraint of 10$^{-2}$
cm in knowing the difference between the semimajor axes $a$ of the satellites,
and the difference $I$ of the inclinations $i$ of the orbital planes which, for
$i\sim 0.01^\circ$, should be less than $0.006^\circ$. A pair of spacecrafts
endowed with a sophisticated intersatellite tracking apparatus and drag-free
control down to 10$^{-9}$ cm s$^{-2}$ Hz$^{-{1/2}}$ level might allow to meet
the stringent requirements posed by such a mission.

General Relativity and Quantum Cosmology;Astrophysics