  The unipotent variety of a reductive algebraic group $G$ plays an important
role in the representation theory. In this paper, we will consider the closure
$\bar{\Cal U}$ of the unipotent variety in the De Concini-Procesi
compactification $\bar{G}$ of a connected simple algebraic group $G$. We will
prove that $\bar{\Cal U}-\Cal U$ is a union of some $G$-stable pieces
introduced by Lusztig in \cite{L4}. This was first conjectured by Lusztig. We
will also give an explicit description of $\bar{\Cal U}$. It turns out that
similar results hold for the closure of any Steinberg fiber in $\bar{G}$.
