A search is conducted for a new scalar boson $S$, with a mass distinct from that of the Higgs boson, decaying promptly into four leptons ($\ell =$ $e$, $\mu$) via an intermediate state containing two on-shell, promptly decaying new spin-1 bosons $Z_{d}$: $S \rightarrow Z_{d}Z_{d} \rightarrow 4\ell$, where the $Z_{d}$ boson has a mass between 15 and 300 GeV, and the $S$ boson has a mass between either 30 and 115 GeV or 130 and 800 GeV. The search uses proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider with an integrated luminosity of 139 fb$^{-1}$ at a centre-of-mass energy of $\sqrt{s}=13$ TeV. No significant excess above the Standard Model background expectation is observed. Upper limits at 95% confidence level are set on the production cross-section times branching ratio, $\sigma(gg \rightarrow S) \times \mathcal{B}(S\rightarrow Z_{d}Z_{d} \rightarrow 4\ell)$, as a function of the mass of both particles, $S$ and $m_{Z_{d}}$.
