Results of three searches are presented for the production of supersymmetric particles decaying into final states with missing transverse momentum and exactly two isolated leptons, e or mu. The analysis uses a data sample collected during the first half of 2011 that corresponds to a total integrated luminosity of 1 fb(-1) of root s = 7 TeV proton-proton collisions recorded with the ATLAS detector at the Large Hadron Collider. Opposite-sign and same-sign dilepton events are separately studied, with no deviations from the Standard Model expectation observed. Additionally, in opposite-sign events, a search is made for an excess of same-flavour over different-flavour lepton pairs. Effective production cross sections in excess of 9.9 fb for opposite-sign events containing supersymmetric particles with missing transverse momentum greater than 250 GeV are excluded at 95% CL For same-sign events containing supersymmetric particles with missing transverse momentum greater than 100 GeV, effective production cross sections in excess of 14.8 fb are excluded at 95% CL The latter limit is interpreted in a simplified electroweak gaugino production model excluding chargino masses up to 200 GeV, under the assumption that slepton decay is dominant.