  We prove that for any free ergodic probability measure preserving action
$\Gamma \actson (X,\mu)$ of a non-elementary hyperbolic group, or a lattice in
a rank one simple Lie group, the associated group measure space II_1 factor
$L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra,
up to unitary conjugacy.
