Measurements at sqrt(s)= 13.6 TeV of the opposite-sign W boson pair production cross section in proton-proton collisions are presented. The data used in this study were collected with the CMS detector at the CERN LHC in 2022, and correspond to an integrated luminosity of 34.8 fb-1. Events are selected by requiring one electron and one muon of opposite charge. A maximum likelihood fit is performed on signal- and background-enriched data categories defined by the flavour and charge of the leptons, the number of jets, and number of jets originating from b quarks. The overall sensitivity is significantly better than previous results with a similar integrated luminosity. The improvement comes from a more refined control of experimental uncertainties and an improved fit strategy. An inclusive WW production cross section of 125.7 +/- 5.6 pb is measured, in agreement with standard model predictions. Cross sections are also reported in a fiducial region close to that of the detector acceptance, both inclusively and differentially, as a function of the jet multiplicity in the event. For the first time in proton-proton collisions, WW events with zero, one, and at least two jets are studied simultaneously and compared with recent theoretical predictions.