The Virasoro Lie algebra is the non-trivial central extension of the Lie algebra of vector fields on the circle. There is a well-known 1-1 correspondence between the coadjoint orbits in the affine dual of the Virasoro algebra and conjugacy classes in a certain open subset of the universal cover of SL(2,R). I will present recent work with Anton Alekseev, in which we extend the correspondence of orbits to a geometric Morita equivalence, taking into account the Lie-Poisson structure. As a result, one obtains a 1-1 correspondence between (certain) Hamiltonian Virasoro spaces and quasi-Hamiltonian spaces for the universal cover of SL(2,R). Applications include Teichmueller moduli spaces of hyperbolic metrics on surfaces with boundary.