The total cross sections for the elastic electroproduction of ρ and J/ψ mesons for Q2 > 8 GeV2 and (W) ≃ 90 GeV/c2 are measured at HERA with the H1 detector. The measurements are for an integrated electron - proton luminosity of ≃ 3 pb-1. The dependences of the total virtual photon - proton (γ*p) cross sections on Q2, W and the momentum transfer squared to the proton (f), and, for the ρ, the dependence on the polar decay angle (cosθ*), are presented. The J/ψ : ρ cross section ratio is determined. The results are discussed in the light of theoretical models and of the interplay of hard and soft physics processes.