  Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and
suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of
$Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove
that any irreducible $T$-module $W$ is both thin and dual thin in the sense of
Terwilliger. We produce two bases for $W$ and describe the action of $T$ on
these bases. We prove that the isomorphism class of $W$ as a $T$-module is
determined by two parameters, the dual endpoint and diameter of $W$. We find a
recurrence which gives the multiplicities with which the irreducible
$T$-modules occur in the standard module. We compute this multiplicity for
those irreducible $T$-modules which have diameter at least $D-3$.
