This thesis presents a search for a heavy neutral Higgs boson, $A$, decaying into a $Z$ boson and another heavy Higgs boson, $H$, using a data sample corresponding to an integrated luminosity, initially, of 36.5 fb$^{-1}$ and, subsequently, increased to 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. 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 95\% confidence-level upper limits on $\sigma(A) \times BR(A \to ZH) \times BR(H \to bb)$ are set. The upper limits are in the range 14--830 fb and 0.0062--0.380 pb for the data sample corresponding to an integrated luminosity of 36.5 fb$^{-1}$ and 139 fb$^{-1}$, respectively. An interpretation of the results in the context of two-Higgs-doublet models is also given.