  We present a calculation of the fully differential cross section for Higgs
boson production in the gluon fusion channel through next-to-next-to-leading
order in perturbative QCD. We apply the method introduced in
\cite{Anastasiou:2003gr} to compute double real emission corrections. Our
calculation permits arbitrary cuts on the final state in the reaction $hh \to H
+ X$. It can be easily extended to include decays of the Higgs boson into
observable final states. In this Letter, we discuss the most important features
of the calculation, and present some examples of physical applications that
illustrate the range of observables that can be studied using our result. We
compute the NNLO rapidity distribution of the Higgs boson, and also calculate
the NNLO rapidity distribution with a veto on jet activity.
