  We prove that for any free ergodic nonsingular nonamenable action \Gamma\
\actson (X,\mu) of all \Gamma\ in a large class of groups including all
hyperbolic groups, the associated group measure space factor $L^\infty(X)
\rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary
conjugacy. This generalizes the probability measure preserving case that was
established in [PV12]. We also prove primeness and indecomposability results
for such crossed products, for the corresponding orbit equivalence relations
and for arbitrary amalgamated free products $M_1 *_B M_2$ over a subalgebra B
of type I.
