The Gromov–Witten theory of Deligne–Mumford stacks is a recent development, and hardly any computations have been done beyond three-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack $\mathbb{P}^2_{D,2}$. Here $D$ is a smooth plane curve and $\mathbb{P}^2_{D,2}$ is locally isomorphic to the stack quotient $[U/(\mathbb{Z}/(2))]$, where $U\to V\subseteq \mathbb{P}^2$ is a double cover branched along $D\cap V$. The introduction discusses an enumerative application of these invariants.