The elastic electroproduction of ρ mesons is studied at HERA with the H1 detector for a photon virtuality in the range 1 < Q2 < 60 GeV2 and for a hadronic centre of mass energy in the range 30 < W < 140 GeV. The shape of the ( π π ) mass distribution in the ρ resonance region is measured as a function of Q2. The full set of ρ spin density matrix elements is determined, and evidence is found for a helicity flip amplitude at the level of 8 +/- 3% of the non-flip amplitudes. Measurements are presented of the dependence of the cross section on Q2, W and t (the four-momentum transfer squared to the proton). They suggest that, especially at large Q2, the γ *p cross section develops a stronger W dependence than that expected from the behaviour of elastic and total hadron-hadron cross sections.