  A semiregular permutation group on a set $\Ome$ is called {\em bi-regular} if
it has two orbits. A classification is given of quasiprimitive permutation
groups with a biregular dihedral subgroup. This is then used to characterize
the family of arc-transitive graphs whose automorphism groups containing a
bi-regular dihedral subgroup. We first show that every such graph is a normal
$r$-cover of an arc-transitive graph whose automorphism group is either
quasiprimitive or bi-quasiprimitive on its vertices, and then classify all such
quasiprimitive or bi-quasiprimitive arc-transitive graphs.
