The production of opposite-charge $W$-boson pairs in proton-proton collisions at $\sqrt{s}$ = 13 TeV is measured using data corresponding to 3.16 fb$^{-1}$ of integrated luminosity collected by the ATLAS detector at the CERN Large Hadron Collider in 2015. Candidate $W$-boson pairs are selected by identifying their leptonic decays into an electron, a muon and neutrinos. Events with reconstructed jets are not included in the candidate event sample. The cross-section measurement is performed in a fiducial phase space close to the experimental acceptance and is compared to theoretical predictions. Agreement is found between the measurement and the most accurate calculations available.