The production of a Z boson in association with a J/ψ meson in proton–proton collisions probes the production mechanisms of quarkonium and heavy flavour in association with vector bosons, and allows studies of multiple parton scattering. Using 20.3fb-1 of data collected with the ATLAS experiment at the LHC in pp collisions at √s = 8 TeV , the first measurement of associated Z+J/ψ production is presented for both prompt and non-prompt J/ψ production, with both signatures having a significance in excess of 5σ . The inclusive production cross-sections for Z boson production (analysed in μ+μ- or e+e- decay modes) in association with prompt and non-prompt J/ψ(→μ+μ-) are measured relative to the inclusive production rate of Z bosons in the same fiducial volume to be (36.8±6.7±2.5)×10-7 and (65.8±9.2±4.2)×10-7 respectively. Normalised differential production cross-section ratios are also determined as a function of the Jψ transverse momentum. The fraction of signal events arising from single and double parton scattering is estimated, and a lower limit of 5.3(3.7)mb at 68(95)% confidence level is placed on the effective cross-section regulating double parton interactions.