Ths is a basic question on Dedekind domains.
Let $R$ be a Dedekind domain, $P$ a non-zero prime ideal of $R$. I know that the localization $R_P$ is a PID, but is it true that the completion $\hat{R}_P$ wrt the valuation $v_P$ induced by $P$ is also a PID?
A proof (or a counterexample) or references will be gratefully accepted.