A measurement of the production cross section of top quark pairs (ttbar) in proton-proton collisions at a center-of-mass energy of 7 TeV recorded with the ATLAS detector at the Large Hadron Collider is reported. Candidate events are selected in the dilepton topology with large missing transverse energy and at least two jets. Using a data sample corresponding to an integrated luminosity of 35 pb^-1, a ttbar production cross section of 171 +/- 20(stat.) +/- 14(syst.) +8-6(lum.) pb is measured for an assumed top quark mass of 172.5 GeV. A second measurement requiring at least one jet identified as coming from a b quark yields a comparable result, demonstrating that the dilepton final states are consistent with being accompanied by b-quark jets. These measurements are in good agreement with Standard Model predictions.