  Consider a divisor D with simple normal crossings in a compact K\"ahler
manifold X. We show in this article that a K\"ahler metric in an arbitrary
class, with constant scalar curvature and cusp singularities along the divisor
is unique in this class when K[D] is ample. This we do by generalizing Chen's
construction of approximate geodesics in the space of K\"ahler metrics, and
proving an approximate version of the Calabi-Yau theorem, both independently of
the ampleness of K[D].
