  We calculate the contributions from QED processes involving two virtual
photons to the cross sections for e^+ e^- annihilation into two charmonium
states with the same C-parity. Generically, the cross sections are three orders
of magnitude smaller than those for charmonia with opposite C-parity because
they are suppressed by a factor of alpha^2/alpha_s^2. However, if both
charmonia have quantum numbers J^{PC} = 1^{--}, then there is a contribution to
the cross section that involves the fragmentation of each photon into a
charmonium. The fragmentation contribution is enhanced by powers of
E_{beam}/m_c, the ratio of the beam energy to the charm-quark mass, and this
enhancement can compensate for the suppression factor that is associated with
the coupling constants. In particular, the predicted cross section for J/psi +
J/psi at the B factories is larger than that for J/psi + eta_c.
