  Consider the ring $R:=\Q[\tau,\tau^{-1}]$ of Laurent polynomials in the
variable $\tau$. The Artin's Pure Braid Groups (or Generalized Pure Braid
Groups) act over $R,$ where the action of every standard generator is the
multiplication by $\tau$. In this paper we consider the cohomology of such
groups with coefficients in the module $R$ (it is well known that such
cohomology is strictly related to the untwisted integral cohomology of the
Milnor fibration naturally associated to the reflection arrangement). We give a
sort of \textit{stability} theorem for the cohomologies of the infinite series
$A$, $B$ and $D,$ finding that these cohomologies stabilize, with respect to
the natural inclusion, at some number of copies of the trivial $R$-module $\Q$.
We also give a formula which compute this number of copies.
