  Let $G$ be a connected, simple algebraic group over an algebraically closed
field. There is a partition of the wonderful compactification $\bar{G}$ of $G$
into finite many $G$-stable pieces, which were introduced by Lusztig. In this
paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We
will show that the closure is a disjoint union of some $G$-stable pieces, which
was first conjectured by Lusztig. We will also prove the existence of cellular
decomposition if the closure contains finitely many $G$-orbits.
