Let $(R, \mathfrak m)$ be an excellent, local normal domain of dimension $2$ (hence Cohen-Macaulay) with an algebraically closed residue field $k=R/\mathfrak m$. Assume that $IJ$ is an integrally closed ideal whenever $I,J$ are integrally closed ideals of $R$ ( https://en.m.wikipedia.org/wiki/Integral_closure_of_an_ideal ). (To see that this purely Algebraic condition on integral closure of ideals is same as saying that the local ring $R$ has rational singularity, see Theorem 1 of https://doi.org/10.1007/BF01233425 )
Then, how to show that the divisor class group $Cl(R)$ is finite ?