A measurement of the correlations between the polar angles of leptons from the decay of pair-produced $t$ and $\overline{t}$ quarks in the helicity basis is reported, using proton-proton collision data collected by the ATLAS detector at the LHC. The dataset corresponds to an integrated luminosity of $4.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at a center-of-mass energy of $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ collected during 2011. Candidate events are selected in the dilepton topology with large missing transverse momentum and at least two jets. The angles ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$ between the charged leptons and the direction of motion of the parent quarks in the $t\overline{t}$ rest frame are sensitive to the spin information, and the distribution of $\mathrm{cos}{\ensuremath{\theta}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{cos}{\ensuremath{\theta}}_{2}$ is sensitive to the spin correlation between the $t$ and $\overline{t}$ quarks. The distribution is unfolded to parton level and compared to the next-to-leading order prediction. A good agreement is observed.