  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 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 GeV
for the $A$ boson and 130-700 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 $\sigma \times B(A\rightarrow ZH) \times
B(H\rightarrow bb$ or $H\rightarrow WW)$ are set. The upper limits are in the
range 0.0062-0.380 pb for the $H\rightarrow bb$ channel and in the range
0.023-8.9 pb for the $H\rightarrow WW$ channel. An interpretation of the
results in the context of two-Higgs-Doublet models is also given.
