This Letter presents a measurement of the W+ W- production cross section in sqrt(s) = 7 TeV pp collisions by the ATLAS experiment, using 34 pb(-1) of integrated luminosity produced by the Large Hadron Collider at CERN. Selecting events with two isolated leptons, each either an electron or a muon, 8 candidate events are observed with an expected background of 1.7 ± 0.6 events. The measured cross section is 41(-16)(+20)(stat) ± 5(syst)±1(lumi) pb, which is consistent with the standard model prediction of 44 ± 3 pb calculated at next-to-leading order in QCD.