  Given a closed surface endowed with a volume form, we equip the space of
compatible Riemannian structures with the structure of an infinite-dimensional
symplectic manifold. We show that the natural action of the group of
volume-preserving diffeomorphisms by push-forward has a group-valued momentum
map that assigns to a Riemannian metric the canonical bundle. We then deduce
that the Teichm\"uller space and the moduli space of Riemann surfaces can be
realized as symplectic orbit reduced spaces.
