  We calculate the lowest order QCD amplitude, i.e. the quark exchange
contribution, to the forward production amplitude of a pair of longitudinally
polarized $\rho$ mesons in the scattering of two virtual photons $\gamma^*(Q_1)
\gamma^*(Q_2) \to \rho^0_L \rho^0_L$. We show that the scattering amplitude
simultaneously factorizes in two quite different ways: the part with transverse
photons is described by the QCD factorization formula involving the generalized
distribution amplitude of two final $\rho$ mesons, whereas the part with
longitudinally polarized photons takes the QCD factorized form with the
  $\gamma^*_L \to \rho^0_L$ transition distribution amplitude. Perturbative
expressions for these, in general, non-perturbative functions are obtained in
terms of the $\rho-$meson distribution amplitude.
