Assume that $A$ is an integrally closed integral domain, and $K$ is its fraction field. Well...this may be a stupid question, but is every overring of $A$ between $A$ and $K$ also integrally closed ? (This is known to be true if $A$ is a Dedekind domain, see e.g. Jarden, Field Arithmetic, chap. 2).
Let $A=K+YK[X,Y]$. We have $Q(A)=K(X,Y)$. To show that $A$ is integrally closed notice that $A\subset K[X,Y]$, and the last ring is integrally closed. If an element $z∈Q(A)$ is integral over $A$, then it is integral over $K[X,Y]$, so it belongs to $K[X,Y]$. Now one can get rid of the part of $z$ which belongs to $A$ and get that $z=f(X)$ is integral over $A$ hence over $K$, so $\deg f=0$.
Now consider $R=K[X^2]+YK[X,Y]$. Clearly $R$ is not integrally closed since $X$ is integral over $R$ and $X\notin R$.