Densest packings of two translates of a lattice in five dimensions

We prove that no packing of unit balls in $\R^5$ whose centres form two translates of a lattice is denser than the $D_5$ lattice packing. Up to congruence exactly three such packings attain this density: $\sqrt2\,D_5$, a layered packing of Leech, and the packing with Szöllősi's kissing configuration.