My question: Let $R$ be an integral domain which is finitely generated as an algebra over a field or $\mathbb{Z}$. Is it true that the completion of $R$ w.r.t. a regular maximal ideal $\mathfrak{m}$ (i.e. the localization $R_\mathfrak{m}$ is a regular local ring) is again an integral domain? (If yes, does this proof exist in the literature somewhere?)
According to this question, this is equivalent to the fact that there is a unique maximal ideal $\mathfrak{M}$ above $\mathfrak{m}$ in the integral closure of $R$ in its quotient field. But I do not know how to prove that in case of regular maximal $\mathfrak{m} \subseteq R$. But other approaches are, of course, welcome as well.