Many high J pure rotational transitions of (HeH+)-He-4 up to the dissociation threshold have been measured by infrared diode laser absorption spectroscopy between 590 and 920 cm(-1) in an air cooled H-2/He discharge. Additional centrifugal distortion terms up to J(12) have been included in the rotational Hamiltonian to fit these transitions. Ten new vibration rotational transitions have also been detected in the same spectral region, arising from both low (v=1,2) and high (v=4-7) vibrational states. Some of the transitions involving high v (v=6,7) are observed as stimulated emission signals. The measured rotational transitions are in excellent agreement with ab initio calculations based on the form of the adiabatic potential model developed by Fournier and Richard.