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.