  We consider the Dirichlet problem for stationary biharmonic maps $u$ from a
bounded, smooth domain $\Omega\subset\mathbb R^n$ ($n\ge 5$) to a compact,
smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any
smooth boundary data, we show that if, in addition, $u$ satisfies a certain
boundary monotonicity inequality, then there exists a closed subset
$\Sigma\subset\bar{\Omega}$, with $H^{n-4}(\Sigma)=0$, such that $u\in
C^\infty(\bar\Omega\setminus\Sigma, N)$.
