  In this paper we introduce the moduli stack $\widetilde{\mathscr{M}}_{g,n}$
of $n$-marked stable at most cuspidal curves of genus $g$ and we use it to
determine the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$. Along the
way, we also determine the integral Chow ring of
$\overline{\mathscr{M}}_{1,2}$.
