  We construct a locally compact Hausdorff topology on the path space of a
directed graph $E$, and identify its boundary-path space $\partial E$ as the
spectrum of a commutative $C^*$-subalgebra $D_E$ of $C^*(E)$. We then show that
$\partial E$ is homeomorphic to a subset of the infinite-path space of any
desingularisation $F$ of $E$. Drinen and Tomforde showed that we can realise
$C^*(E)$ as a full corner of $C^*(F)$, and we deduce that $D_E$ is isomorphic
to a corner of $D_F$. Lastly, we show that this isomorphism implements the
homeomorphism between the boundary-path spaces.
