I am looking for an integral domain $A$ with the following properties:


1. $A$ is not integrally closed
2. $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
3. There is an integral element $x\in K$ (*over* $A$) such that $A[x]$ is integrally closed.


Can someone help to tell me if the above is even possible? 


**Edit**: Lubin easily gave me an example. Now I want to consider the case when I replace the condition 2. by:


2'. $A$ has a quotient field $K$ that is real closed.



Non-normal domain with algebraically closed fraction field
Try this: Let $B\_0$ be the ring of real algebraic integers, and let $B=B\_0[1/2]$, so the ring of real algebraic numbers integral except possibly at $2$. But $B[i]$ is equal to the ring of algebraic numbers integral except possibly at $2$, and this is integrally closed. And so we take $A=B[3i]$, not integrally closed, and of course the fraction field is algebraically closed.


