The inclusive top-quark-pair production cross section rrtt and its ratio to the Z-boson production cross section have been measured in proton-proton collisions at root s = 13.6 TeV, using 29 fb-1 of data collected in 2022 with the ATLAS experiment at the Large Hadron Collider. Using events with an opposite-charge electron-muon pair and b-tagged jets, and assuming Standard Model decays, the top-quark-pair production cross section is measured to be rrtt = 850 +/- 3(stat.) +/- 18(syst.) +/- 20(lumi.) pb. The ratio of the ttand the Z-boson production cross sections is also measured, where the Z-boson contribution is determined for inclusive e+e- and mu+mu- events in a fiducial phase space. The relative uncertainty on the ratio is reduced compared to the ttcross section, thanks to the cancellation of several systematic uncertainties. The result for the ratio, Rtt/Z = 1.145 +/- 0.003(stat.) +/- 0.021(syst.) +/- 0.002(lumi.) is consistent with the Standard Model prediction using the PDF4LHC21 PDF set.