This Letter presents a measurement of the $\WW$ production cross section in $\sqrt{s} = 7 \TeV\ pp$ collisions by the ATLAS experiment, using 34 $\pbi$ 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 $1.7\pm0.6$ events. The measured cross section is $41^{+20}_{-16}(\mathrm{stat.})\pm 5(\mathrm{syst.})\pm 1(\mathrm{lumi.})$ pb, which is consistent with the standard model prediction of $44 \pm 3$ pb calculated at next-to-leading order in QCD.