  Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field
${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of
ideals defining $X$. Assume that the set of regular points of $X$ is dense in
$X$. We prove that there exists a proper, birational morphism, $\pi:
W_r\longrightarrow W$, obtained as a composition of monoidal transformations,
so that if $X_r\subset W_r$ denotes the strict transform of $X\subset W$ then:
1) The morphism $\pi:W_r\longrightarrow W$ is an embedded desingularization of
$X$ (as in Hironaka's Theorem); 2) The {\em total transform} of ${\mathcal
I}(X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of
ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of
ideals defining the strict transform of $X$ (i.e. ${\mathcal I}(X){\mathcal
O}_{W_r}={\mathcal L}\cdot{\mathcal I}(X_r)$). This result is stronger than
Hironaka's Theorem, in fact (2) is novel and does not hold for
desingularizations which follow Hironaka's line of proof unless $X$ is a
hypersurface. We will say that $W_r\longrightarrow W$ defines a {\em Strong
Desingularization of $X$}.
