Cross sections for elastic production of J/φ mesons in photoproduction and electroproduction are measured in electron proton collisions at HERA using an integrated luminosity of 55 pb-1. Results are presented for photon virtualities Q2 up to 80 GeV2. The dependence on the photon-proton centre of mass energy Wγp is analysed in the range 40 ≤ Wγp ≤ 305 GeV in photoproduction and 40 ≤ Wγp ≤ 160 GeV in electroproduction. The Wγp dependences of the cross sections do not change significantly with Q 2 and can be described by models based on perturbative QCD. Within such models, the data show a high sensitivity to the gluon density of the proton in the domain of low Bjorken x and low Q2. Differential cross sections dσ/dt, where t is the squared four-momentum transfer at the proton vertex, are measured in the range |t| < 1.2 GeV2 as functions of Wγp and Q2. Effective Pomeron trajectories are determined for photoproduction and electroproduction. The J/φ production and decay angular distributions are consistent with s-channel helicity conservation. The ratio of the cross sections for longitudinally and transversely polarised photons is measured as a function of Q2 and is found to be described by perturbative QCD based models.