We study antipodal distance-regular graphs of diameter 3 such that their group of automorphisms acts transitively on the set of pairs , where is an edge of the graph. Hence the group of automorphisms of the graph acts 2-transitively on the set of antipodal classes, so the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices with distance at most two have exactly common neighbors.