  A longstanding conjecture on biharmonic submanifolds, proposed by Chen in
1991, is that {\it any biharmonic submanifold in a Euclidean space is minimal}.
In the case of a hypersurface $M^n$ in $\mathbb R^{n+1}$, Chen's conjecture was
settled in the case of $n=2$ by Chen and Jiang around 1987 independently.
Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with
$n=3$. However, the general Chen's conjecture on a hypersurface $M^n$ remains
open for $n> 3$. In this paper, we settle Chen's conjecture for hypersurfaces
in $\mathbb R^{5}$ for $n=4$.
