  We study the limit in low intensity of Poisson--Voronoi tessellations in
hyperbolic spaces $ \mathbb{H}_{d}$ for $d \geq 2$. In contrast to the
Euclidean setting, a limiting nontrivial ideal tessellation $ \mathcal{V}_{d}$
appears as the intensity tends to $0$. The tessellation $ \mathcal{V}_{d}$ is a
natural, isometry-invariant decomposition of $ \mathbb{H}_{d}$ into countably
many unbounded polytopes, each with a unique end. We study its basic
properties, in particular, the geometric features of its cells.
