We give a proof of the Singer conjecture (on the vanishing of reduced
$\ell^2$-homology except in the middle dimension) for the Davis Complex
$\Sigma$ associated to a Coxeter system $(W,S)$ whose nerve $L$ is a
triangulation of $\mathbb{S}^2$. We show that it follows from a theorem of
Andreev, which gives the necessary and sufficient conditions for a classical
reflection group to act on $\mathbb{H}^3$.