  We obtain a necessary and sufficient condition of existence of a
K{\"a}hler-Einstein metric on a $G\times G$-equivariant Fano compactification
of a complex connected reductive group $G$ in terms of the associated polytope.
This condition is not equivalent to the vanishing of the Futaki invariant. The
proof relies on the continuity method and its translation into a real
Monge-Amp{\`e}re equation, using the invariance under the action of a maximal
compact subgroup $K\times K$.
