More formal: Let a normal algebraic singularity be given by a local ring $R$ of finite type. Is there always an affine variety $X$ with a point $x$, so that

- $\widehat{\mathcal{O}}_{X,x} \cong \widehat{R}$
- $CL(X) \cong CL(R)$ or equivalently $Pic(X)=0$

where $CL$ and $Pic$ denote the (Weil) divisor class group and the Picard group respectively. This seems to be a very natural question to me, but I found nothing on it. I would be very happy about any ideas or references.