  The exclusive production of pairs of vector mesons with J^{PC} = 1^{--} in
e^+ e^- collisions can proceed through e^+ e^- annihilation into two virtual
photons. At energies much greater than the meson masses, the cross section is
dominated by the independent fragmentation of the virtual photons into the
vector mesons. The fragmentation approximation is used to calculate the cross
sections and angular distributions for pairs of vector mesons that can be
produced at the B factories. The predicted cross sections for rho^0 + rho^0 and
rho^0 + phi production agree with recent measurements by the BaBar
Collaboration. For the production of two charmonium vector mesons, the
nonfragmentation corrections to the cross sections are calculated by using the
NRQCD factorization formalism. The predicted cross sections for J/psi + J/psi
and J/psi + psi(2S) production are compatible with upper limits set by the
Belle Collaboration.
