  We present the analytic evaluation of the two-loop corrections to the
amplitude for the scattering of four fermions in Quantum Electrodynamics, $f^-
+ f^+ + F^- + F^+ \to 0$, with $f$ and $F$ representing a massless and a
massive lepton, respectively. Dimensional regularization is employed to
evaluate the loop integrals. Ultraviolet divergences are removed by
renormalizing the coupling constant in the ${\overline{\text{MS}}}$-scheme, and
the lepton mass as well as the external fields in the on-shell scheme. The
analytic result for the renormalized amplitude is expressed as Laurent series
around $d=4$ space-time dimensions, and contains Generalized Polylogarithms
with up to weight four. The structure of the residual infrared divergences of
the virtual amplitude is in agreement with the prediction of the Soft Collinear
Effective Theory. Our analytic results are an essential ingredient for the
computation of the scattering cross section for massive fermion-pair production
in massless fermion-pair annihilation, i.e. $f^- f^+ \to F^- F^+$, and crossing
related processes such as the elastic scattering $f F \to f F$, with up to
Next-to-Next to Leading Order accuracy.
