In the course of reading a paper , I've encountered the following property of interest.

If $R$ is a ring, say it satisfies (*) if: For any smooth, irreducible $R$-algebra $B$ of finite type such that all the fibers of $Spec B$ over points of codimension one in $Spec R$ are irreducible, then $(B \otimes_R K)^* = B^* K^*$, where $K$ is the fraction field of $R$.

The author remarks that it is easy to verify property () for UFDs. However, I don't see how to do this. What's a proof UFDs satisfy ()?

The application I'm interested in is actually where $R$ is a DVR. I feel that in this case, one should be able to give an even simpler argument.