Is the integral closure of a polynomial ring a UFD?

Question

Let $\mathbb{C}(x)$ be the field of rational functions over the complex numbers and $F$ a finite extension of $\mathbb{C}(x)$. Suppose $B$ is the integral closure of $\mathbb{C}[x]$ (the ring of polynomials over $\mathbb{C}$). Is $B$ always a Unique Factorization Domain (UFD)?