It is known that the hyperbolic and parabolic complex are two fundamental types for infinite and simply connected complexes

whose corresponding circle packings fill the hyperbolic and the Euclidean plane

respectively. Given an infinite simply connected complex K with boundary

it is proved that there exists an univalent circle packing P for K in the hyperbolic plane D whose circles associate with boundary vertices of K are horocycles

which is complete in the sense of permitting the existence of interstices between horocycles and unit circle D. Moreover

the circle packing P is unique up to Mbius transformations of D.