This Letter presents a measurement of the W þ W À production cross section in ffiffi ffi s p ¼ 7 TeV pp collisions by the ATLAS experiment, using 34 pb À1 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 AE 0:6 events. The measured cross section is 41 þ20 À16 ðstatÞ AE 5ðsystÞ AE 1ðlumiÞ pb, which is consistent with the standard model prediction of 44 AE 3 pb calculated at next-to-leading order in QCD.