Abstract: The production of a Z boson in association with a J/psi 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 20.3 fb(-1) of data collected with theATLAS experiment at the LHC in pp collisions at root s = 8 TeV, the first measurement of associated Z + J/psi production is presented for both prompt and non-prompt J/psi production, with both signatures having a significance in excess of 5 sigma. The inclusive production cross-sections for Z boson production (analysed in mu(+) mu(-) or e(+) e(-) decay modes) in association with prompt and non-prompt J/psi (-> mu(+) mu(-)) are measured relative to the inclusive production rate of Z bosons in the same fiducial volume to be (36.8 +/- 6.7 +/- 2.5) x 10(-7) and (65.8 +/- 9.2 +/- 4.2) x 10(-7) respectively. Normalised differential production cross-section ratios are also determined as a function of the J/psi 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.