  The two-loop (Euler-Heisenberg-type) effective action for N = 2
supersymmetric QED is computed using the N = 1 superspace formulation. The
effective action is expressed as a series in supersymmetric extensions of
F^{2n}, where n=2,3,..., with F the field strength. The corresponding
coefficients are given by triple proper-time integrals which are evaluated
exactly. As a by-product, we demonstrate the appearance of a non-vanishing F^4
quantum correction at the two-loop order. The latter result is in conflict with
the conclusion of hep-th/9710142 that no such quantum corrections are generated
at two loops in generic N = 2 SYM theories on the Coulomb branch. We explain a
subtle loophole in the relevant consideration of hep-th/9710142 and re-derive
the F^4 term from harmonic supergraphs.
