The set of coadjoint orbits of the Virasoro algebra at level 1 is in
bijection with the set of conjugacy classes in a certain open subset
$\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm
SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of
quasi-symplectic groupoids, integrating the Poisson structure on
$\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on
$\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.
