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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.