  This paper reports cross-section measurements of $ZZ$ production in $pp$
collisions at $\sqrt{s}=13.6$ TeV at the Large Hadron Collider. The data were
collected by the ATLAS detector in 2022, and correspond to an integrated
luminosity of 29 fb$^-1$. Events in the $ZZ\rightarrow4\ell$ ($\ell = e$,
$\mu$) final states are selected and used to measure the inclusive and
differential cross-sections in a fiducial region defined close to the analysis
selections. The inclusive cross-section is further extrapolated to the total
phase space with a requirement of 66 $< m_Z <$ 116 GeV for both $Z$ bosons,
yielding $16.8 \pm 1.1$ pb. The results are well described by the Standard
Model predictions.
