  We study the analyticity properties of the free energy $f_\ga(m)$ of the Kac
model at points of first order phase transition, in the van der Waals limit
$\ga\searrow 0$. We show that there exists an inverse temperature $\beta_0$ and
$\ga_0>0$ such that for all $\beta\geq \beta_0$ and for all $\ga\in(0,\ga_0)$,
$f_\ga(m)$ has no analytic continuation along the path $m\searrow m^*$ ($m^*$
denotes spontaneous magnetization). The proof consists in studying high order
derivatives of the pressure $p_\ga(h)$, which is related to the free energy
$f_\ga(m)$ by a Legendre transform.
