  Two planar embedded circle patterns with the same combinatorics and the same
intersection angles can be considered to define a discrete conformal map. We
show that two locally finite circle patterns covering the unit disc are related
by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement
for the complex plane if all exterior intersection angles of neighboring
circles are uniformly bounded away from $0$.
  Finally, we study a sequence of two circle patterns with the same
combinatorics each of which approximates a given simply connected domain.
Assume that all kites are convex and all angles in the kites are uniformly
bounded and the radii of one circle pattern converge to $0$. Then a subsequence
of the corresponding discrete conformal maps converges to a Riemann map between
the given domains.
