A measurement of the production of a prompt $J/\psi$ meson in association with a $W\pm$ boson with $W\pm \to \mu \nu$ and $J/\psi \to \mu +\mu -$ is presented for $J/\psi$ transverse momenta in the range 8.5-150 GeV and rapidity 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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The ratio of the prompt $J/\psi$ plus $W\pm$ cross-section to the inclusive $W\pm$ cross-section is presented as a differential measurement as a function of $J/\psi$ transverse momenta and compared with theoretical predictions using different double-parton-scattering cross-sections.