  Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In
this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$
and the maximal generating degree $d(I)$ of $I$ such that $\depth
R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an
application, we classify the class of monomial ideals $I$ such that
$\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.
