Let $X$ be a compact manifold with boundary $\partial X$, and suppose
that $\partial X$ is the total space of a fibration
\[
Z\rightarrow \partial X \rightarrow Y\, .
\]
Let $D_\Phi$ be a generalized Dirac operator associated
to a $\Phi$-metric $g_\Phi$ on $X$. Under the assumption that
$D_\Phi$ is fully elliptic we prove an index formula for $D_\Phi$.
The proof is in two steps: first, using results of Melrose and Rochon,
we show that the index is unchanged if we pass to a certain
$b$-metric $g_b (\epsilon)$. Next we write the $b-$ (i.e. the APS) index formula
for $g_b(\ep)$; the $\Phi$-index formula follows by analyzing the limiting behaviour
as $\epsilon\searrow 0$ of the two terms in the formula. The interior term
is studied directly whereas the adiabatic limit formula for the eta invariant
follows from work of Bismut and Cheeger.