  Let $K/k$ be a finite abelian CM-extension and $T$ a suitable finite set of
finite primes of $k$. In this paper, we determine the Fitting ideal of the
minus component of the $T$-ray class group of $K$, except for the
$2$-component, assuming the validity of the equivariant Tamagawa number
conjecture. As an application, we give a necessary and sufficient condition for
the Stickelberger element to lie in that Fitting ideal.
