Results are presented of searches for the production of supersymmetric particles decaying into final states with missing transverse momentum and exactly two isolated leptons in sqrt{s}=7 TeV proton-proton collisions at the Large Hadron Collider. Search strategies requiring lepton pairs with identical sign or opposite sign electric charges are described. In a data sample corresponding to an integrated luminosity of 35 pb-1 collected with the ATLAS detector, no significant excesses are observed. Based on specific benchmark models, limits are placed on the squark mass between 450 and 690 GeV for squarks approximately degenerate in mass with gluinos, depending on the supersymmetric mass hierarchy considered. Show Partial Abstract