  In this paper we determine the structure of the Chow ring of the
Delaunay-Voronoi compactification $\tilde{\cal A}_3$ of the moduli space of
principally polarized abelian threefolds. This compactification was introduced
by Namikawa and studied by Tsushima. We use equivariant classes on level
coverings of $\tilde{\cal A}_3$. We also compare this ring with the Chow ring
of the moduli space of stable genus 3 curves as determined by Faber.
