The requirement that the supersymmetric scalar potential be stable in the minimal supergravity (mSUGRA) model imposes an upper bound on the universal gaugino mass m1/2 as function of the common scalar mass m0. Combining this with the experimental lower bound on m1/2 from LEP data, we find a new lower bound on m0, stronger than the one that comes from experimental data alone. If the corresponding upper limits on the superparticle masses, derived in this Letter, are found to be violated at Tevatron Run II or at the LHC, it would imply that we are living on a false vacuum. Special attention has been paid in estimating the uncertainties in these predictions due to the choice of the renormalization scale. The implications of our limits for the constraints obtained by indirect methods (SUSY dark matter, g-2 of the muon, b→sγ...) are briefly discussed. © 2004 Elsevier B.V. All rights reserved.