  We show that the dipole-dipole interaction between two Rydberg atoms can give
rise to long range molecules. The binding potential arises from two states that
converge to different separated atom asymptotes. These states interact weakly
at large distances, but start to repel each other strongly as the van der Waals
interaction turns into a resonant dipole-dipole interaction with decreasing
separation between the atoms. This mechanism leads to the formation of an
attractive well for one of the potentials. If the two separated atom asymptotes
come from the small Stark splitting of an atomic Rydberg level, which lifts the
Zeeman degeneracy, the depth of the well and the location of its minimum are
controlled by the external electric field. We discuss two different geometries
that result in a localized and a donut shaped potential, respectively.
