Results on J/ψ production in ep interactions in the H1 experiment at HERA are presented. The J/ψ mesons are produced by almost real photons (Q2 ≈ 0) and detected via their leptonic decays. The data have been taken in 1994 and correspond to an integrated luminosity of 2.7 pb-1. The γp cross section for elastic J/ψ production is observed to increase strongly with the center of mass energy. The cross section for diffractive J/ψ production with proton dissociation is found to be of similar magnitude as the elastic cross section. Distributions of transverse momentum and decay angle are studied and found to be in accord with a diffractive production mechanism. For inelastic J/ψ production the total γp cross section, the distribution of transverse momenta, and the elasticity of the J/ψ are compared to NLO QCD calculations in a colour singlet model and agreement is found. Diffractive ψ' production has been observed and a first estimate of the ratio to J/ψ production in the HERA energy regime is given.