  Let $\Gamma$ denote a distance-regular graph with diameter $D \ge 3$. Assume
$\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $b < -1$. Let $X$
denote the vertex set of $\Gamma$ and let $A \in MX$ denote the adjacency
matrix of $\Gamma$. Fix $x \in X$ and let $A^* \in MX$ denote the corresponding
dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A,
A^*$. We call $T$ the {\em Terwilliger algebra} of $\Gamma$ with respect to
$x$. We show that up to isomorphism there exist exactly two irreducible
$T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these
$T$-modules we display a basis consisting of eigenvectors for $A^*$, and for
each basis we give the action of $A$
