  We give an example of a set $\Omega \subset \R^5$ which is a finite union of
unit cubes, such that $L^2(\Omega)$ admits an orthonormal basis of exponentials
$\{\frac{1}{|\Omega|^{1/2}} e^{2\pi i \xi_j \cdot x}: \xi_j \in \Lambda \}$ for
some discrete set $\Lambda \subset \R^5$, but which does not tile $\R^5$ by
translations. This answers a conjecture of Fuglede in the negative, at least in
5 and higher dimensions.
