The rotational spectra of the Ne-20 and Ne-22 isotopomers of the Ne-dimethyl sulfide (DMS) rare gas dimer have been measured by Fourier transform microwave spectroscopy. MP2/6-311++G(2d, 2p) calculations, and the experimental spectroscopic data, suggest a structure of C-s symmetry in which the Ne atom lies above the heavy atom plane of the DMS ( in the sigma v plane which bisects the CSC angle). Experimental rotational constants are consistent with a S center dot center dot center dot Ne distance of 3.943(6) angstrom and a (cm center dot center dot center dot S center dot center dot center dot Ne) angle of 63.2(6)degrees (where cm is the center of mass of DMS). A motion of the Ne atom from one side of the DMS to the other gives rise to inversion splittings of around 3 MHz in the c-type transitions. An ab initio potential energy surface calculation has allowed examination of several possible tunneling pathways, and suggests a barrier of between 20 and 40 cm(-1) for the inversion motion, depending on the tunneling pathway taken by the Ne. Dipole moment measurements are consistent with both the experimental and ab initio structures.