  We establish a criterion for determining when a smooth Deligne-Mumford stack
is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack
$\mathcal{X}$ and a Cartier divisor $\mathcal{E} \subset \mathcal{X}$ such that
(1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne-Mumford
stack $\mathcal{Y}$ and (2) for every $y\in\mathcal{Y}$ we have
$\mathcal{O}_{\mathcal{X}}(\mathcal{E})|_{\mathcal{E}_y}\simeq
\mathcal{O}_{\mathcal{E}_y}(-1)$, then there exists a contraction
$\mathcal{X}\to\mathcal{Z}$ to a smooth Deligne-Mumford stack $\mathcal{Z}$.
Moreover, the stack $\mathcal{X}$ can be recovered as a weighted blow-up along
$\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$, and
$\mathcal{Z}$ is a pushout in the category of algebraic stacks. As an
application, we show that the moduli stack $\overline{\mathscr{M}}_{1,n}$ of
stable $n$-pointed genus one curves is a weighted blow-up of the stack of
pseudo-stable curves. Along the way we also prove a reconstruction result for
smooth Deligne-Mumford stacks that is of independent interest.
