Is the blowup of an integral normal Noetherian scheme along a coherent sheaf of ideals necessarily normal?

I can show that there is an open cover of the blowup by schemes of the form $\text{Spec } C$, where $B \subset C \subset B_g$ for some integrally closed domain $B$ and some $g \in B$, but I don't see why this would imply that $C$ is integrally closed. Intuitively, it seems reasonable that a blowup would be at least as "nice" as the original scheme, but that intuition may have more to do with how blowups are generally used than what they are capable of.

everybirational map between projective varieties over a field is a blow-up along some closed subscheme of the target. General blow-ups are totally wild things. – BCnrd Sep 4 '10 at 19:27