This Letter reports on a search for narrow high-mass resonances decaying into dilepton final states. The data were recorded by the ATLAS experiment in pp collisions at sqrt(s) = 7 TeV at the LHC and correspond to a total integrated luminosity of 1.08/fb (1.21/fb) in the ee (mumu) channel. No statistically significant excess above the Standard Model expectation is observed and upper limits are set at the 95% C.L. on the cross section times branching ratio of Z' resonances and Randall-Sundrum gravitons decaying into dileptons as a function of the resonance mass. A lower mass limit of 1.83 TeV on the Sequential Standard Model Z' boson is set. A Randall-Sundrum graviton with coupling k/M_Pl=0.1 is excluded at 95% C.L. for masses below 1.63 TeV. Show Partial Abstract