  For a non-exceptional oriented surface S let Q(S) be the moduli space of area
one quadratic differentials. We show that there is a Borel subset E of Q(S)
which is invariant under the Teichmueller flow F^t and of full measure for
every invariant Borel probability measure, and there is a measurable conjugacy
of the restriction of F^t to E into the Weil-Petersson flow. This conjugacy
induces a continuous injection H of the space of invariant Borel probability
measures for F^t into the space of invariant Borel probability measures for the
Weil-Petersson flow. The map H is not surjective, but its image contains the
Lebesgue Liouville measure.
