  Let the warped product $M^n=L^m\times_\varphi F^{n-m}$, $n\geq m+3\geq 8$, of
Riemannian manifolds be an Einstein manifold with Ricci curvature $\rho$ that
admits an isometric immersion into Euclidean space with codimension two. Under
the assumption that $L^m$ is also Einstein, but not of constant sectional
curvature, it is shown that $\rho=0$ and that the submanifold is locally a
cylinder with an Euclidean factor of dimension at least $n-m$. Hence $L^m$ is
also Ricci flat. If $M^n$ is complete, then the same conclusion holds globally
if the assumption on $L^m$ is replaced by the much weaker condition that either
its scalar curvature $S_L$ is constant or that $S_L\leq (2m-n)\rho$.
