I have a ring $R=k[S]$ which defines an affine toric variety $X_\sigma$, where $S=M\cap \sigma^\vee$ is the semigroup from a rational polyhedral cone $\sigma$. Let $I$ be the ideal for the toric boundary divisor of $X_\sigma$, in other words, $I$ is obtained from the toric ideal of $S$ that is $>0$ on the closure of $\sigma$. I want to know if the $I$-adic completion $\hat{R}$ of $R$ is still an integral domain?

A simple example is $R=k[x,y]$, and $I$ is the principal ideal $(xy)$. Is the completion an integral domain?