Interactions of heavy Majorana neutrinos in the thermal phase of the early universe may be the origin of the cosmological matter-antimatter asymmetry. This mechanism of baryogenesis implies stringent constraints on light and heavy Majorana neutrino masses. We derive an improved upper bound on the /CP asymmetry in heavy neutrino decays which, together with the kinetic equations, yields an upper bound on all light neutrino masses of 0.1 eV. Lepton number changing processes at temperatures above the temperature TB of baryogenesis can erase other, pre-existing contributions to the baryon asymmetry. We find that these washout processes become very efficient if the effective neutrino mass m~1 is larger than m*~=10-3 eV. All memory of the initial conditions is then erased. Hence, for neutrino masses in the range from √Δm2sol~=8×10-3 eV to √Δm2atm~=5×10-2 eV, which is suggested by neutrino oscillations, leptogenesis emerges as the unique source of the cosmological matter-antimatter asymmetry.