Let $G$ be a connected semisimple linear algebraic group over an algebraically closed field $k$ of positive characteristic and let $X$ denote an equivariant embedding of $G$. We define a distinguished Steinberg fiber $N$ in $G$, called the zero-fiber, and prove that the closure of $N$ within $X$ is normal and Cohen-Macaulay. Furthermore, when $X$ is smooth we prove that the closure of $N$ is a local complete intersection.