The production of two prompt J/psi mesons, each with transverse momenta p(T) > 8.5 GeV and rapidity vertical bar y vertical bar < 2.1, is studied using a sample of proton-proton collisions at root 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/psi production, is measured as a function of the transverse momentum of the lower-p(T) J/psi meson, di-J/psi p(T) and mass, the difference in rapidity between the two J/psi mesons, and the azimuthal angle between the two J/psi mesons. The fraction of prompt pair events due to double parton scattering is determined by studying kinematic correlations between the two J/psi 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 sigma(eff) = 6.3 +/- 1.6(stat) +/- 1.0(syst) mb.