Very accurate variational calculations of the five lowest vibrational states of the (HeHe+)-He-3-He-4 ion are carried out within a framework that does not assume the Born-Oppenheimer (BO) approximation, i.e., treating the two nuclei and three electrons forming the system on an equal footing. The non-BO wave functions are expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. The wave functions are used to calculate the leading relativistic corrections. The approach reproduces the experimental (HeHe+)-He-3-He-4 fundamental transition within 0.055 cm(-1) and similar accuracy is expected for the higher yet unmeasured vibrational transitions determined in the present calculations. (C) 2010 Elsevier B.V. All rights reserved.