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.