  In the context of pure QED, we obtain analytic expressions for the
contributions to the Bhabha scattering differential cross section at order
alpha^4 which originate from the interference of two-loop photonic vertices
with tree-level diagrams and from the interference of one-loop photonic
diagrams amongst themselves. The ultraviolet renormalization is carried out.
The IR-divergent soft-photon emission corrections are evaluated and added to
the virtual cross section. The cross section obtained in this manner is valid
for on-shell electrons and positrons of finite mass, and for arbitrary values
of the center of mass energy and momentum transfer. We provide the expansion of
our results in powers of the electron mass, and we compare them with the
corresponding expansion of the complete order alpha^4 photonic cross section,
recently obtained in hep-ph/0501120. As a by-product, we obtain the
contribution to the Bhabha scattering differential cross section of the
interference of the two-loop photonic boxes with the tree-level diagrams, up to
terms suppressed by positive powers of the electron mass. We evaluate
numerically the various contributions to the cross section, paying particular
attention to the comparison between exact and expanded results.
