Normalization and factor ring

Hi folks !

My work is stuck on this question :

Let A be a domain (a quotient of a polynomial algebra over a field is enough).

Consider A' its normalization, and p a prime ideal of A.

Suppose that A/p is integrally closed, then is A'/(pA') integrally closed ?

The question can be geometrically rephrased as :

Let X an algebraic variety, and X' its normalization. Is the pull-back of a normal variety normal again ?

I already know, thanks to Bourbaki, that the components of the pull-back do not interset each other, but I don't know how to say more, even considering the special case where the sub-variety is a smooth curve.

I think that the answer is affirmative.

Thank you very much for any idea you would have !