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 t pairs and of their subsequent decays. In this Letter we present for the reactions p p, pp→tt+X →ℓ+ℓ−+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 tt state. The angular distribution reflects the degree of correlation of the t and t spins which we determine for different choices of t and 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.