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We prove that for any free ergodic probability measure preserving actionΓ  →  (  X  ,  μ  )    $\Gamma \rightarrow (X,\mu )$    of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II  1   factorL   ∞    (  X  )   ⋊  Γ    ${\mathord {\text{\rm L}}^\infty (X) \rtimes \Gamma }$    hasL   ∞    (  X  )${\mathord {\text{\rm L}}^\infty (X)}$    as its unique Cartan subalgebra, up to unitary conjugacy.

Unique Cartan decomposition for II 1 factors arising from arbitrary actions of hyperbolic groups