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.
