The diffractive photoproduction of J/ψ mesons is measured with the H1 detector at the ep collider HERA using an integrated luminosity of 78 pb−1. The differential cross section dσ(γp→J/ψY)/dt is studied in the range 2<|t|<30 GeV2, where t is the square of the four-momentum transferred at the proton vertex. The cross section is also presented as a function of the photon–proton centre-of-mass energy Wγp in three t intervals, spanning the range 50<Wγp<200 GeV. A fast rise of the cross section with Wγp is observed for each t range and the slope for the effective linear Pomeron trajectory is measured to be ′=−0.0135±0.0074(stat.)±0.0051(syst.) GeV−2. The measurements are compared with perturbative QCD models based on BFKL and DGLAP evolution. The data are found to be compatible with s-channel helicity conservation.