The reaction γ p → J/ψ p has been studied in ep interactions using the ZEUS detector at HERA. The cross section for elastic J/ψ photoproduction has been measured as a function of the photon-proton centre of mass energy W in the range 40 < W < 140 GeV at a median photon virtuality Q2 of 5 × 10-5 GeV2. The photoproduction cross section, σγp→J/ψp, is observed to rise steeply with W. A fit to the data presented in this paper to determine the parameter δ in the form σγp→J/ψp α Wδ yields the value δ = 0.92±0.14±0.10. The differential cross section dσ/d|t| is presented over the range |t| < 1.0 GeV2 where t is the square of the four-momentum exchanged at the proton vertex. dσ/d|t| falls exponentially with a slope parameter of 4.6 ± 0.4-0.6+0.4 GeV-2. The measured decay angular distributions are consistent with s-channel helicity conservation. © Springer-Verlag 1997.