  In this paper we shall establish that properly embedded constant mean
curvature one surfaces in H^3 of finite topology are of finite total curvature
and each end is regular. In particular, this implies the horosphere is the only
simply connected such example, and the catenoid cousins the only annular
examples of this nature. In general each annular end of such a surface is
asymptotic to an end of a horosphere or an end of a catenoid cousin.
