The production of J/ψ mesons in proton-proton collisions at √S = 7 TeV is studied with the LHCb detector at the LHC. The differential cross-section for prompt J/ψ production is measured as a function of the J/ψ transverse momentum pT and rapidity y in the fiducial region pT ∈ [0; 14] GeV/c and y ∈ [2.0; 4.5]. The differential cross-section and fraction of J/ψ from b-hadron decays are also measured in the same pT and y ranges. The analysis is based on a data sample corresponding to an integrated luminosity of 5.2 pb^&-1. The measured cross-sections integrated over the fiducial region are 10.52 ± 0.04 ± 1.40^+1.64 &-2.20 μb for prompt J/ψ production and 1.14 ± 0.01 ± 0.16 μb for J/ψ from b-hadron decays, where the first uncertainty is statistical and the second systematic. The prompt J/ψ production cross-section is obtained assuming no J/ψ polarisation and the third error indicates the acceptance uncertainty due to this assumption.