The production of two longitudinally polarized $Z$ bosons is measured using proton–proton collision data recorded by the ATLAS experiment at a center-of-mass energy $\sqrt{s}=13.6$ TeV and with an integrated luminosity of 164 fb${}^{-1}$. The $ZZ$ candidate events are reconstructed using two same-flavor opposite-charge electron or muon pairs. The polarization fractions of the $Z$ boson pair are measured inclusively in the fiducial phase space and also in two regions of the four-lepton invariant mass, and the results are consistent with the Standard Model predictions. After a statistical combination with a previous measurement at $\sqrt{s}=13$ TeV, the first observation of the production of two simultaneously longitudinally polarized $Z$ bosons is obtained, with observed and expected significances of 6.5 and 5.6 standard deviations, respectively.