All literature on vibration-rotational and pure rotational transition energies for the X1Sigma+ state of 4HeH+, 3HeH+, 4HeD+, and 3HeD+ have been employed in a direct least-squares fit of the radially dependent Hamiltonian. The Born-Oppenheimer potential is represented by a modified Lennard-Jones function that provides for correct behavior in the near dissociation long-range region. Only 19 adjustable parameters were required to represent the 166 measured transition energies, which are reproduced almost to within the measurement accuracies. The fitted potential and Born-Oppenheimer breakdown functions are shown to be in good agreement with the results of ab initio calculations. The results are used to obtain high-order centrifugal distortion constants as well as level widths for quasi-bound levels. Copyright 1999 Academic Press.