  In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$
come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H >
1$. Via the Lawson correspondence the latter two cases correspond to constant
mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H
\neq 0$, respectively. These surface classes have been investigated intensively
in the literature. For the case $0 \leq H < 1$ there is no Lawson
correspondence in Euclidean space and there are relatively few publications.
Examples have been difficult to construct. In this paper we present a
generalized Weierstra{\ss} type representation for surfaces of constant mean
curvature in $\mathbb{H}^3$ with particular emphasis on the case of mean
curvature $0\leq H < 1$. In particular, the generalized Weierstra{\ss} type
representation presented in this paper enables us to construct simultaneously
minimal surfaces (H=0) and non-minimal constant mean curvature surfaces
($0<H<1$).
