The production of two prompt $J/\psi$ mesons, each with transverse momenta $p_{\mathrm{T}}>8.5$ GeV and rapidity $|y| < 2.1$, is studied using a sample of proton-proton collisions at $\sqrt{s} = 8$ 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-$p_{\mathrm{T}}$ $J/\psi$ meson, di-$J/\psi$ $p_{\mathrm{T}}$ 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 $\sigma_{\mathrm{eff}} = 6.3 \pm 1.6 \mathrm{(stat)} \pm 1.0 \mathrm{(syst)}$ mb.