  Let $G/H$ be a compact homogeneous space, and let $\hat{g}_0$ and $\hat{g}_1$
be $G$-invariant Riemannian metrics on $G/H$. We consider the problem of
finding a $G$-invariant Einstein metric $g$ on the manifold $G/H\times [0,1]$
subject to the constraint that $g$ restricted to $G/H\times \{0\}$ and
$G/H\times \{1\}$ coincides with $\hat{g}_0$ and $\hat{g}_1$, respectively. By
assuming that the isotropy representation of $G/H$ consists of pairwise
inequivalent irreducible summands, we show that we can always find such an
Einstein metric.
