  Let $\mathcal{X} \to Y$ be a birational modification of a variety by an Artin
stack. In previous work, under the assumption that $\mathcal{X}$ is smooth, we
proved a change of variables formula relating motivic integrals over arcs of
$Y$ to motivic integrals over arcs of $\mathcal{X}$. In this paper, we extend
that result to the case where $\mathcal{X}$ is singular. We may therefore apply
this generalized formula to the so-called warping stack
$\mathscr{W}(\mathcal{X})$ of $\mathcal{X}$, which may be singular even when
$\mathcal{X}$ is smooth. We thus obtain a change of variables formula
\emph{canonically} expressing any given motivic integral over arcs of $Y$ as a
motivic integral over \emph{warped arcs} of $\mathcal{X}$.
