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 and the possibility of measuring it is examined. The observable is the difference of the times required for l to pass from 0 to 2π for both senses of motion. Such a gravitomagnetic time shift, which is independent of the orbital parameters of the satellites, amounts to 5×10⁻⁷ s for the Earth; it is cumulative and should be measurable 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 ~ 0.01°, should be less than 0.006°. A pair of spacecraft endowed with a sophisticated intersatellite tracking apparatus and drag-free control down to 10^-9 cm s^-2 Hz^-1/2 level might allow us to meet the stringent requirements posed by such a mission.