  We give a solution to the inverse problem of Moebius geometry on the circle.
Namely, we describe a class of Moebius structures on the circle for each of
which there is a hyperbolic space such that its boundary at infinity is the
circle, and the induced Moebius structure coincides with the given one. That
class is not empty and form an open neighborhood of the canonical Moebius
structure in an appropriate fine topology.
