  Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus
two. For a slope $\gamma$ on $F$, we denote by $M(\gamma)$ the manifold
obtained by attaching a 2-handle to $M$ along a regular neighborhood of
$\gamma$ on $F$. In this paper, we shall prove that there is at most one
separating slope $\gamma$ on $F$ so that $M(\gamma)$ is $\partial$-reducible.
