A measurement of the production of a prompt J/ψ meson in association with a W ± boson with W ± → μν and J/ψ → μ +μ − is presented for J/ψ transverse momenta in the range 8.5–150 GeV and rapidity |y J/ψ| <2.1 using ATLAS data recorded in 2012 at the LHC. The data were taken at a proton-proton centre-of-mass energy of s = 8 TeV and correspond to an integrated luminosity of 20.3 fb −1. The ratio of the prompt J/ψ plus W ± cross-section to the inclusive W ± cross-section is presented as a differential measurement as a function of J/ψ transverse momenta and compared with theoretical predictions using different double-parton-scattering cross-sections. [Figure not available: see fulltext.].