  We consider cross-product II$_1$ factors $M = N\rtimes_{\sigma} G$, with $G$
discrete ICC groups that contain infinite normal subgroups with the relative
property (T) and $\sigma: G \to {\text{\rm Aut}}N$ trace preserving actions of
$G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing.
Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity
result for such factors, showing the uniqueness of the position of $L(G)$
inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in
terms of the weights of the shift, for certain arithmetic groups $G$ such as
$G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. We deduce that for any countable group $S
\subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus
bringing new light to a longstanding problem of Murray and von Neumann.
