Cross sections for elastic and proton-dissociative photoproduction of J/ψ mesons are measured with the H1 detector in positron-proton collisions at HERA. The data were collected at ep centre-of-mass energies √s ≈ 318 GeV and √s ≈ 225 GeV, corresponding to integrated luminosities of L = 130 pb−1 and L = 10.8 pb−1, respectively. The cross sections are measured as a function of the photon-proton centre-of-mass energy in the range 25 < Wγp < 110 GeV. 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 for the elastic process and |t| < 8 GeV2 for proton dissociation. The results are compared to other measurements. The Wγp and t-dependences are parametrised using phenomenological fits.