  We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M}
= \Gamma(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of
$\R$ on a separable real Hilbert space $H_\R$, the continuous core $M =
\mathcal{M} \rtimes_\sigma \R$ is a semisolid ${\rm II_\infty}$ factor, i.e.
for any non-zero finite projection $q \in M$, the ${\rm II_1}$ factor $qMq$ is
semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then
we prove that the core $M$ is solid. As an application, we construct an example
of a non-amenable solid ${\rm II_1}$ factor $N$ with full fundamental group,
i.e. $\mathcal{F}(N) = \R^*_+$, which is not isomorphic to any interpolated
free group factor $L(\F_t)$, for $1 < t \leq +\infty$.
