  Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and
let $u_\lambda$ be a Dirichlet Laplace eigenfunction in $\Omega$ with
eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of
the zero set of $u_\lambda$ does not exceed $C(\Omega)\sqrt{\lambda}$. This
result is new even for the case of domains with $C^\infty$-smooth boundary.
