Ideal contraction rigidity under an integral extension

Let $A\subseteq B$ be integral and let $I\subseteq J$ be ideals of $B$. If $I$ is prime and $I\cap A=J\cap A$, then $I=J$. After quotienting by $I$, a nonzero element of $J/I$ has an integral equation of least degree whose nonzero constant term lies in $(J/I)\cap (A/(I\cap A))$, a contradiction.