  This paper is devoted to the two-loop QED radiative corrections to the decay
pi0 -> e+ e-. We compute the virtual corrections without using any
approximation and we take into account all the relevant graphs with the
inclusion of those omitted in the previous approximative calculations. The
bremsstrahlung is then treated within the soft photon approximation. We
concentrate on the technical aspects of the calculation and discuss in detail
the UV renormalization and the treatment of IR divergences within the
dimensional regularization. As a result we obtain the O(alpha^3 p^2)
contribution in closed analytic form. We compare the exact two-loop results
with existing approximative calculations of QED corrections and find
significant disagreement in the kinematical region relevant for the KTeV
experiment.
