Ever since Galileo scientists have known that all bodies fall with the same acceleration regardless of their mass and composition. Known as the Universality of Free Fall, this is the most direct experimental evidence of the Weak Equivalence Principle, a founding pillar of General Relativity according to which the gravitational (passive) mass mg and the inertial mass mi are always in the same positive ratio in all test bodies. A space experiment offers two main advantages: a signal about a factor of a thousand bigger than on Earth and the absence of weight. A new space mission named GALILEO GALILEI (GG) has been proposed (Nobili et al., 1995 [J. Astronautical Sciences, 43, 219]; GALILEO GALILEI (GG), PRE PHASE A REPORT, ASI (Agenzia Spaziale Italiana), September 1996) aimed at testing the weak Equivalence Principle (EP) to 1 part in 10 17 in a rapidly spinning (5 Hz) drag-free spacecraft at room temperature, the most recent ground experiments having reached the level of 10 -12 (Adelberger et al., 1990 [PhRvD, 42, 3267]; Su et al., 1994 [PhRvD, 50, 3614]). Here we present a nondrag-free version of GG which could reach a sensitivity of 1 part in 10 16. The main feature of GG is that, similarly to the most recent ground experiments, the expected (low frequency) signal is modulated at higher frequency by spinning the system, in this case by rotating the test bodies (in the shape of hollow cylinders) around their symmetry axes, the signal being in the perpendicular plane. They are mechanically suspended inside the spacecraft and have very low frequencies of natural oscillation (due to the weakness of the springs that can be used because of weightlessness) so as to allow self-centering of the axes; vibrational noise around the spin/signal frequency is attenuated by means of mechanical suspensions. The signal of an EP violation would appear at the spin frequency as a relative ( differential) displacement of the test masses perpendicularly to the spin axis, and be detected by capacitance sensors; thermal stability across the test masses and for the required integration time is obtained passively thanks to both the fast spin and the cylindrical symmetry. In the nondrag-free version the entire effect of atmospheric drag is retained, but a very accurate balancing of the test bodies must be ensured (through a coupled suspension) so as to reach a high level of Common Mode Rejection and reduce the differential effects of drag below the target sensitivity. In so doing the complexities of a drag-free spacecraft are avoided by putting more stringent requirements on the experiment. The spacecraft must have a high area-to-mass ratio in order to reduce the effects of nongravitational forces; it is therefore a natural choice to have three pairs of test masses (in three experimental chambers) rather than one as by Nobili et al. (1995) [J. Astronautical Sciences, 43, 219] and the mission called GALILEO GALILEI [PRE PHASE A REPORT, ASI (Agenzia Spaziale Italiana), September 1996]. The GG setup is specifically designed for space; however, a significant EP test on the ground is possible — because the signal is in the transverse plane — by exploiting the horizontal component of the gravitational and the centrifugal field of the Earth. This ground test is underway.