Exclusive ρ0ρ0 production in two-photon collisions involving a single highly virtual photon is studied with data collected at LEP at centre-of-mass energies 89 GeV < √s < 209 GeV with a total integrated luminosity of 854.7 pb−1. The cross section of the process γ γ ∗ → ρ0ρ0 is determined as a function of the photon virtuality, Q2, and the two-photon centre-of-mass energy, Wγ γ , in the kinematic region: 1.2 GeV2 < Q2 < 30 GeV2 and 1.1 GeV < Wγ γ < 3 GeV.