  Positronium annihilation amplitudes that are computed by assuming a
factorization approximation with on-shell intermediate leptons, do not exhibit
good analytical behavior. We propose an ansatz which allows to include binding
energy corrections and obtain the correct analytical and gauge invariance
behavior of these QED amplitudes. As a consequence of these non-perturbative
corrections, the parapositronium and orthopositronium decay rates receive
corrections of order alpha^4 and alpha^2, respectively. These new corrections
for orthopositronium are relevant in view of a precise comparison between
recent theoretical and experimental developments. Implications are pointed out
for analogous decays of quarkonia .
