  The inclusive decay B --> X_u l nu is of much interest because of its
potential to constrain the CKM element |V_ub|. Experimental cuts required to
suppress charm background restrict measurements of this decay to the
shape-function region, where the hadronic final state carries a large energy
but only a moderate invariant mass. In this kinematic region, the differential
decay distributions satisfy a factorization formula of the form $H \cdot J
\otimes S$, where S is the non-perturbative shape function, and the object $H
\cdot J$ is a perturbatively calculable hard-scattering kernel. In this paper
we present the calculation of the hard function H at next-to-next-to-leading
order (NNLO) in perturbation theory. Combined with the known NNLO result for
the jet function J, this completes the perturbative part of the NNLO
calculation for this process.
