A measurement of spin correlation in ttbar production is reported using data collected with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 2.1 fb^-1. Candidate events are selected in the dilepton topology with large missing transverse energy and at least two jets. The difference in azimuthal angle between the two charged leptons in the laboratory frame is used to extract the correlation between the top and antitop quark spins. In the helicity basis the measured degree of correlation corresponds to 0.40 +0.09 -0.08, in agreement with the next-to-leading-order Standard Model prediction. The hypothesis of zero spin correlation is excluded at 5.1 standard deviations.