  The collider experiments at the Tevatron and the LHC will allow for detailed
investigations of the properties of the top quark. This requires precise
predictions of the hadronic production of $t\bar t$ pairs and of their
subsequent decays. In this Letter we present for the reactions $p {\bar p}, p p
\to t{\bar t} + X \to \ell^+\ell'^-X$ the first calculation of the dilepton
angular distribution at next-to-leading order (NLO) in the QCD coupling,
keeping the full dependence on the spins of the intermediate $t\bar{t}$ state.
The angular distribution reflects the degree of correlation of the $t$ and
$\bar t$ spins which we determine for different choices of $t$ and $\bar t$
spin bases. In the case of the Tevatron, the QCD corrections are sizeable, and
the distribution is quite sensitive to the parton content of the proton.
