The production of two prompt J/ψ mesons, each with transverse momenta pT>8.5 GeV and rapidity |y|<2.1, is studied using a sample of proton-proton collisions at √s=8 TeV, corresponding to an integrated luminosity of 11.4 fb−1 collected in 2012 with the ATLAS detector at the LHC. The differential cross-section, assuming unpolarised J/ψ production, is measured as a function of the transverse momentum of the lower-pT J/ψ meson, di-J/ψpT and mass, the difference in rapidity between the two J/ψ mesons, and the azimuthal angle between the two J/ψ mesons. The fraction of prompt pair events due to double parton scattering is determined by studying kinematic correlations between the two J/ψ mesons. The total and double parton scattering cross-sections are compared with predictions. The effective cross-section of double parton scattering is measured to be σeff=6.3±1.6(stat)±1.0(syst) mb.