A search for a heavy neutral Higgs boson, A, decaying into a Z boson and another heavy Higgs boson, H, is performed using a data sample corresponding to an integrated luminosity of 139 fb^{-1} from proton–proton collisions at sqrt{s} = 13 text {TeV} recorded by the ATLAS detector at the LHC. The search considers the Z boson decaying into electrons or muons and the H boson into a pair of b-quarks or W bosons. The mass range considered is 230–800 text {GeV} for the A boson and 130–700 text {GeV} for the H boson. The data are in good agreement with the background predicted by the Standard Model, and therefore 95% confidence-level upper limits for sigma times B(Arightarrow ZH)times B(Hrightarrow bb ; text {or} ; H rightarrow WW) are set. The upper limits are in the range 0.0062–0.380 pb for the Hrightarrow bb channel and in the range 0.023–8.9 pb for the Hrightarrow WW channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.