The production of J/ψ mesons is studied in pp collisions at $\sqrt{s} = 7$ TeV with the CMS experiment at the LHC. The measurement is based on a dimuon sample corresponding to an integrated luminosity of 314 nb−1. The J/ψ differential cross section is determined, as a function of the J/ψ transverse momentum, in three rapidity ranges. A fit to the decay length distribution is used to separate the prompt from the non-prompt (b hadron to J/ψ) component. Integrated over J/ψ transverse momentum from 6.5 to 30 GeV/c and over rapidity in the range |y|<2.4, the measured cross sections, times the dimuon decay branching fraction, are 70.9±2.1(stat.)±3.0(syst.)±7.8(luminosity) nb for prompt J/ψ mesons assuming unpolarized production and 26.0±1.4(stat.)±1.6(syst.)±2.9(luminosity) nb for J/ψ mesons from b-hadron decays.