  The positronium molecule (Ps$_2$) has not been experimentally observed yet
because its tiny (4.5 eV) binding energy cannot be detected when the molecule
annihilates by emitting two photons with energy of 0.51 MeV each. It is shown
in this paper that the electric dipole transition between the recently found
L=1 excited-state and the L=0 ground-state with its characteristic photon
energy of 4.94 eV is a clear signature of the existence of the positronium
molecule and the possibility of its experimental observation is realistic. The
probability of this transition is about 17 % of the total decay rate. An other
Coulomb four-body system containing positron, HPs (the positronium hydride or
hydrogen positride), is also included for comparison.
