  We prove an existence result for non rotational constant mean curvature ends
in $\mathbb{H}^2 \times \mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic
real plane. The value of the curvature is $h \, \in \, (0, 1/2)$. We use
Schauder theory and a continuity method for solution of the prescribed mean
curvature equation of exterior domains of $\mathbb{H}^2$. We also prove a fine
property of the asymptotic behavior of the rotational ends introduced by Sa
Earp and Toubiana.
