  We present an estimate of the cross-section for the exclusive production of a
rho_L^0-meson pair in e^+e^- scattering, which will be studied in the future
high energy International Linear Collider. For this aim, we complete
calculations of the Born order approximation of the amplitudes
gamma^*_{L,T}(Q_1^2) gamma^*_{L,T}(Q_2^2) -> rho_L^0 rho_L^0, for arbitrary
polarization of virtual photons and longitudinally polarized mesons, in the
kinematical region s >> -t, Q_1^2, Q_2^2. These processes are completely
calculable in the hard region Q_1^2, Q_2^2 >> Lambda^2_{QCD} and we perform
most of the calculations in an analytical way. The resulting cross-section
turns out to be large enough for this process to be measurable with foreseen
luminosity and energy, for Q_1^2 and Q_2^2 in the range of a few GeV^2.
