We argue that the observed antiproton production in heavy-ion collisions at CERN-SPS energies can be understood if (contrary to most sequential scattering approaches) the backward direction in the process pP<-->n(pi) (with n = 5-6) is consistently accounted for within a thermal framework. Employing the standard picture of subsequent chemical and thermal freezeout, which induces an oversaturation of pion number with associated chemical potentials of mu(pi) approximately 60-80 MeV, enhances the backward reaction substantially. The resulting rates turn out to be large enough to maintain an antiproton abundance at thermal freezeout in accordance with the measured p/p ratio in Pb(158A GeV)+Pb collisions.