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 83 GeV⩽s⩽209 GeV with a total integrated luminosity of 684.8 pb−1. The cross section of the γγ∗→ρ+ρ− process is determined as a function of the photon virtuality, Q2Q2, and the two-photon centre-of-mass energy, WγγWγγ, in the kinematic region: 0.2 GeV2⩽Q2⩽0.85 GeV2 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 γγ∗→ρρ process over the Q2-region 0.2 GeV2⩽Q2⩽30 GeV2.