A search for a heavy, CP-odd Higgs boson, A, decaying into a Z boson and a 125 GeV Higgs boson, h, with the ATLAS detector at the LHC is presented. The search uses proton-proton collision data at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 20.3 fb(-1). Decays of CP-even h bosons to tau tau or bb pairs with the Z boson decaying to electron or muon pairs are considered, as well as h -> bb decays with the Z boson decaying to neutrinos. No evidence for the production of an A boson in these channels is found and the 95% confidence level upper limits derived for sigma(gg -> A) x BR(A -> Zh) x BR(h -> f (f) over bar) are 0.098-0.013 pb for f = tau and 0.57-0.014 pb for f = b in a range of m(A) = 220-1000 GeV. The results are combined and interpreted in the context of two-Higgs-doublet models. Published by Elsevier B.V.