  We study the relationship between two sets of coordinates on a double Bruhat
cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and
the $\CX$-coordinates defined by the coweight parametrization of Fock and
Goncharov. In these coordinates, we show that the generalized Chamber Ansatz of
Fomin and Zelevinsky is a nondegenerate version of the canonical monomial
transformation between the cluster variables and $\CX$-coordinates defined by a
common exchange matrix. We prove this in the setting of an arbitrary
symmetrizable Kac-Moody group, generalizing along the way many previous results
on the double Bruhat cells of a semisimple algebraic group. In particular, we
construct an upper cluster algebra structure on the coordinate ring of any
double Bruhat cell in a symmetrizable Kac-Moody group, proving a conjecture of
Berenstein, Fomin, and Zelevinsky.
