  We classify non-complete prime valency graphs satisfying the property that
their automorphism group is transitive on both the set of arcs and the set of
$2$-geodesics. We prove that either $\Gamma$ is 2-arc transitive or the valency
$p$ satisfies $p\equiv 1\pmod 4$, and for each such prime there is a unique
graph with this property: it is a non-bipartite antipodal double cover of the
complete graph $K_{p+1}$ with automorphism group $PSL(2,p)\times Z_2$ and
diameter 3.
