  We show that the Tate-Hochschild cohomology ring $HH^*(RG,RG)$ of a finite
group algebra $RG$ is isomorphic to a direct sum of the Tate cohomology rings
of the centralizers of conjugacy class representatives of $G$. Moreover, our
main result provides an explicit formula for the cup product in $HH^*(RG,RG)$
with respect to this decomposition. As an example, this formula helps us to
compute the Tate-Hochschild cohomology ring of the symmetric group $S_3$ with
coefficients in a field of characteristic 3.
