Double diffractive rho-production in gamma^* gamma^* collisions

25 pages, 5 figures. Published version in EPJC. Minor changes and several references added with respect to v1We present a first estimate of the cross-section for the exclusive process gamma^*_L (Q_1^2) gamma^*_L(Q_2^2) -> rho^0_L rho^0_L, which will be studied in the future high energy e^+ e^- linear collider. As a first step, we calculate the Born order approximation of the amplitude for longitudinally polarized virtual photons and mesons, in the kinematical region s >> -t, Q_1^2, Q_2^2. This process is completely calculable in the hard region Q_1^2, Q_2^2 >> Lambda^2_{QCD}. We perform most of the steps 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