  We study natural cluster structures in the rings of regular functions on
simple complex Lie groups and Poisson-Lie structures compatible with these
cluster structures. According to our main conjecture, each class in the
Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds
to a cluster structure in $\O(\G)$. We have shown before that this conjecture
holds for any $\G$ in the case of the standard Poisson--Lie structure and for
all Belavin-Drinfeld classes in $SL_n$, $n<5$. In this paper we establish it
for the Cremmer-Gervais Poisson-Lie structure on $SL_n$, which is the least
similar to the standard one. Besides, we prove that on $SL_3$ the cluster
algebra and the upper cluster algebra corresponding to the Cremmer-Gervais
cluster structure do not coincide, unlike the case of the standard cluster
structure. Finally, we show that the positive locus with respect to the
  Cremmer-Gervais cluster structure is contained in the set of totally positive
matrices.
