  We show the existence of ultra-long-range giant dipole molecules formed by a
neutral alkali ground state atom that is bound to the decentered electronic
wave function of a giant dipole atom. The adiabatic potential surfaces emerging
from the interaction of the ground state atom with the giant dipole electron
posses a rich topology depending on the degree of electronic excitation.
Binding energies and the vibrational motion in the energetically lowest
surfaces are analyzed by means of perturbation theory and exact diagonalization
techniques. The resulting molecules are truly giant with internuclear distances
up to several $\mu m$. Finally, we demonstrate the existence of intersection
manifolds of excited electronic states that potentially lead to a vibrational
decay of the ground state atom dynamics.
