Relativistic core-potential calculations have been carried out on the ground and excited Omega states of ArHe and HeNe correlating with the atomic limits Ar*(3p(5)4s, P-3,P-1) and Ne*(2p(5)3s, P-3, P-1), respectively, and with ground-state He atoms. The potential energy curves of the excited states of ArHe are repulsive. The potential energy curves of the 0(-)(I), 1(I) and 0(+)(II)2p sigma --> 3s states of HeNe show a local Rydberg minimum at 2.8 bohr and a barrier at 4.0 bohr, while the other excited state potentials of HeNe are repulsive. For both ArHe and HeNe, large dipole transition moments are calculated connecting the ground state with states dissociating to atomic limits from which radiative transitions to the atomic ground state are allowed. Small but nonzero dipole transition moments are obtained for short internuclear distances for the transitions from the 1(I) state, which correlates with the metastable P-3(2) state of Ar (in ArHe) or Ne (in HeNe). The radiative lifetime of the 1(I) state of HeNe (estimated from the vertical transition at 2.8 bohr) is 18 mus.