  The production and annihilation of antiprotons in the hadronic phase of
heavy-ion collisions is evaluated within a thermal equilibrium approach. It is
shown that the inverse reaction of $p\bar p$ annihilation (i.e., multi-pion
annihilation $N_\pi \pi\to p\bar p$ with $N_\pi\simeq$~5-7) in connection with
oversaturation of pion phase space (i.e., finite pion chemical potentials)
plays an important role in understanding the observed antiproton yields at SpS
energies within the standard picture of subsequent chemical and thermal
freezeout. Implications for RHIC energies are also addressed.
