  We consider a compact manifold whose boundary is a locally trivial fiber
bundle and an associated pseudodifferential algebra that models fibered cusps
at infinity. Using trace-like functionals that generate the 0-dimensional
Hochschild cohomology groups, we express the index of a fully elliptic fibered
cusp operator as the sum of a local contribution from the interior and a term
that comes from the boundary. This answers the index problem formulated by
Mazzeo and Melrose. We give a more precise answer in the case where the base of
the boundary fiber bundle is the circle. In particular, for Dirac operators
associated to a "product fibered cusp metric", the index is given by the
integral of the Atiyah-Singer form in the interior minus the adiabatic limit of
the eta invariant of the restriction of the operator to the boundary.
