  The elastic electroproduction of rho mesons is studied at HERA with the H1
detector for a photon virtuality in the range 1 < Q^2 < 60 GeV^2 and for a
hadronic centre of mass energy in the range 30 < W < 140 GeV. The shape of the
pipi mass distribution in the rho resonance region is measured as a function of
Q^2. The full set of rho 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 Q^2, W and t (the four-momentum transfer squared to the proton).
They suggest that, especially at large Q^2, the gamma^*p cross section develops
a stronger W dependence than that expected from the behaviour of elastic and
total hadron-hadron cross sections.
