The exclusive photoproduction of J/ψ mesons, γp → J/ψp, has been studied in ep collisions with the ZEUS detector at HERA, in the kinematic range 20 < W < 290 GeV, where W is the photon-proton centre-of-mass energy. The J/ψ mesons were reconstructed in the ninon and the electron decay channels using integrated luminosities of 38 pb-1 and 55 pb-1, respectively. The helicity structure of J/ψ production shows that the hypothesis of s-channel helicity conservation is satisfied within two standard deviations. The total cross section and the differential cross-section dσ/dt, where t is the squared four-momentum transfer at the proton vertex, are presented as a function of W, for |t| < 18 Ge2. The t distribution exhibits an exponential shape with a slope parameter increasing logarithmically with W with a value b = 4.15 ± 0.05(stat.)-0.18+0.30(syst.) GeV-2 at W = 90 GeV. The effective parameters of the Pomeron trajectory are αℙ(0) = 1.200 ± 0.009(stat.) -0.010+0.004(Syst.) and α′ℙ = 0.115 ± 0.018(stat.)-0.015+0.008(syst.) GeV-2.