Affine $k$-domain, which is also a PID and whose group of units is same as that of $k$

Question

Let $k$ be an algebraically closed field of characteristic zero. Let $R$ be a finitely generated $k$-algebra such that $R$ is a non-field PID and $R^*=k^*$ , where $R^*$ denotes the group of units of $R$. Then is it true that $R=k[f]$ for some $f \in k[X]$ i.e. $R$ is isomorphic to $k[X]$ as $k$-algebras ?