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.

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