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 TeX and TeX , corresponding to integrated luminosities of TeX and TeX , respectively. The cross sections are measured as a function of the photon-proton centre-of-mass energy in the range 25<W γp <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 W γp and t-dependences are parametrised using phenomenological fits.