  We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$
and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that
any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined
into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$
factors, with $B$ equal to either $\Bbb C$, to a Cartan subalgebra $A$ in
$M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the
following type of unique decomposition results, \`a la Bass-Serre: If $M = (N_1
*_C N_2 *_C ...)^t$, for some $t>0$ and some other similar inclusions of
algebras $C\subset N_j$ then, after a permutation of indices, $(B\subset M_i)$
is inner conjugate to $(C\subset N_i)^t$, $\forall i$. Taking $B=\Bbb C$ and
$M_i = (L(\Bbb Z^2 \rtimes \Bbb F_{2}))^{t_i}$, with $\{t_i\}_{i\geq 1}=S$ a
given countable subgroup of $\Bbb R_+^*$, we obtain continuously many non
stably isomorphic factors $M$ with fundamental group $\mycal F(M)$ equal to
$S$. For $B=A$, we obtain a new class of factors $M$ with unique Cartan
subalgebra decomposition, with a large subclass satisfying $\mycal F(M)=\{1\}$
and Out$(M)$ abelian and calculable. Taking $B=R$, we get examples of factors
with $\mycal F(M)=\{1\}$, Out$(M)=K$, for any given separable compact abelian
group $K$.
