The inclusive J/ψ production cross-section and fraction of J/ψ mesons produced in B-hadron decays are measured in proton–proton collisions at √s=7 TeV with the ATLAS detector at the LHC, as a function of the transverse momentum and rapidity of the J/ψ, using 2.3 pb−1 of integrated luminosity. The cross-section is measured from a minimum pT of 1 GeV to a maximum of 70 GeV and for rapidities within |y|<2.4 giving the widest reach of any measurement of J/ψ production to date. The differential production cross-sections of prompt and non-prompt J/ψ are separately determined and are compared to Colour Singlet NNLO⋆, Colour Evaporation Model, and FONLL predictions.