Exclusive ρ+ρ− production in two-photon collisions between a quasi-real photon, γ *, and a mid-virtuality photon, γ ∗, is studied with data collected at LEP at centre-of-mass energies 183 GeV ≼ √s ≼ 209 GeV with a total integrated luminosity of 684.8 pb−1. The cross section of the γ γ ∗ → ρ+ρ− process is determined as a function of the photon virtuality, Q2, and the two-photon centre-of-mass energy, Wγγ , in the kinematic region: 0.2 GeV2 ≼ Q2 ≼ 0.85 GeV2 and 1.1 GeV ≼ Wγγ ≼ 3 GeV. These results, together with previous L3 measurements of ρ0ρ0 and ρ+ρ− production, allow a study of the γ γ ∗ → ρρ process over the Q2-region 0.2 GeV2 ≼ Q2 ≼ 30 GeV2.