  Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space
$\mathbb{H}^{n+1}.$ It is proved that if $\Gamma\subset\left\{
x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded
$C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\} $ then,
given $\left\vert H\right\vert <1,$ there is a complete, properly embedded, CMC
$H$ hypersurface $\Sigma$ of $\mathbb{H}^{n+1}$ such that $\partial_{\infty
}S=\Gamma\cup\left\{ x_{n+1}=+\infty\right\} .$ This result can be seen as a
limit case of the existence theorem proved by B. Guan and J. Spruck in
\cite{GS} on CMC $\left\vert H\right\vert <1$ radial graphs with prescribed
$C^{0}$ asymptotic boundary data.
