A measurement is reported of the production cross section of top-quark pairs (tt) in proton-proton collisions at a center-of-mass energy of 7 TeV recorded with the ATLAS detector at the LHC. Candidate events have a signature consistent with containing two isolated leptons, large missing transverse momentum, and at least two jets. Using a data sample corresponding to an integrated luminosity of 0.70 fb−1, a tt production cross section σtt=176±5(stat.)+14−11(syst.)±8(lum.) pb is measured for an assumed top-quark mass of m t  = 172.5 GeV. This measurement is in good agreement with Standard Model predictions.