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 ↔nπ (withn=5-6) is consistently accounted for within a thermal framework. Employing the standard picture of subsequent chemical and thermal freezeout, which induces an over-saturation of pion number with associated chemical potentials of µπ ≃ 60-80 MeV, enhances the backward reaction substantially. The resulting rates and corresponding cross sections turn out to be large enough to maintain the abundance of antiprotons at chemical freezeout until the decoupling temperature, in accord with the measuredp/p ratio in Pb(158AGeV)+Pb collisions.Over the last decade remarkable progress has been made in the understanding of the dynamics of strong interactions probed through (ultra-) relativistic heavy-ion collisions. Although the main challenge of an unambiguous identification of the QCD phase transition to the Quark-Gluon Plasma (QGP) persists, we have greatly advanced our knowledge on properties of highly excited hadronic matter close to the expected phase boundary. A variety of collective phenomena has been observed indicating that the produced systems have indeed reached macroscopically large sizes, justifying the use of equilibrium techniques such as thermo-and hydrodynamics.One of the important results that will be used below is that the final-state hadron abundances, including antibaryons, can be rather accurately characterized by the so-called chemical freezeout stage [1] with a common temperature T ch and baryon chemical potential µ ch B , the specific values depending on collision energy (note that a precise description of all hadron species requires corrections to an ideal gas ensemble, e.g., excluded (eigen-) volumes [1,2] to mimic short-range repulsions, or 'strangeness suppression' factors [2]. Such corrections are not important for our subsequent analysis and will be neglected. For a contrasting view of hadron production in heavy-ion collisions, see, e.g., ref.[3]).At SpS energies, the chemical freezeout is clearly distinct from the thermal one (with an associated temperature T th < T ch ), from where on the particles stream freely to the detectors. This follows from the kinetics of hadronic reactions [4,5], i.e., a significant difference between elastic and inelastic collision rates at low relative energies, and has been confirmed by several experimental evidences (see, e.g., ref.[6]). In central Pb+Pb collisions, e.g., a nucleon at midrapidity is elastically rescattered on average about 10-15 times, but less than once inelastically [7]. Also, as shown in [5], most of the collective flow effects at SpS are generated in between the two freezeouts, and their observation leaves no doubt about the existence of such an intermediate stage.Another consequence is that abundances of secondary mesons (pions, kaons, etc.) are not subject to significant changes when the system evolves from T ch to T th . Again, t...