  Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local
ring and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$
is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let
$G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0
}(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of
the integral closure filtration of $\mathfrak{a}$. Itoh conjectured that if
$e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^*
$ is Cohen-Macaulay. In this paper we prove an important case of Itoh's
conjecture: we show that if $A$ is Cohen-Macaulay and if $\mathfrak{a}$ is
normal (i.e., $\mathfrak{a}^n$ is integrally closed for all $n \geq 1$) with
$e_3^\mathfrak{a}(A) = 0$ then $G_\mathfrak{a}(A)$ is Cohen-Macaulay.
