  We calculate the leading order BFKL amplitude for the exclusive diffractive
process \gamma*_L(Q1^2) \gamma*_L(Q2^2) \to \rho_L^0 \rho_L^0 in the forward
direction, which can be studied in future high energy e^+e^- linear colliders.
The resummation effects are very large compared to the fixed-order calculation.
We also estimate the next-to-leading logarithmic corrections to the amplitude
by using a specific resummation of higher order effects and find a substantial
growth with energy, but smaller than in the leading logarithmic approximation.
