  We prove that every group measure space II$_1$ factor
$L^{\infty}(X)\rtimes\Gamma$ coming from a free ergodic rigid (in the sense of
[Po01]) probability measure preserving action of a group $\Gamma$ with positive
first $\ell^2$--Betti number, has a unique group measure space Cartan
subalgebra, up to unitary conjugacy. We deduce that many $\Cal H\Cal T$
factors, including the II$_1$ factors associated with the actions
$\Gamma\curvearrowright \Bbb T^2$ and $\Gamma\curvearrowright$ SL$_2(\Bbb
R)$/SL$_2(\Bbb Z)$, where $\Gamma$ is a non--amenable subgroup of SL$_2(\Bbb
Z)$, have a unique group measure space Cartan subalgebra.
