We determine the fraction of $t\overline{t}$ events with spin correlation, assuming that the spin of the top quark is either correlated with the spin of the top antiquark as predicted by the standard model or is uncorrelated. For the first time we use a matrix-element-based approach to study $t\overline{t}$ spin correlation. We use $t\overline{t}\ensuremath{\rightarrow}{W}^{+}b{W}^{\ensuremath{-}}\overline{b}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}\ensuremath{\nu}b{\ensuremath{\ell}}^{\ensuremath{-}}\overline{\ensuremath{\nu}}\overline{b}$ final states produced in $p\overline{p}$ collisions at a center-of-mass energy $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$, where $\ensuremath{\ell}$ denotes an electron or a muon. The data correspond to an integrated luminosity of $5.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ and were collected with the D0 detector at the Fermilab Tevatron collider. The result agrees with the standard model prediction. We exclude the hypothesis that the spins of the $t\overline{t}$ are uncorrelated at the 97.7% C.L.