This work reports the multiphoton ionization of cesium atoms, via resonance with the $6F$ level, by interaction with a single transverse and longitudinal mode neodymium-glass laser beam. At a low rate of ion production, when the laser frequency is tuned through resonance, the laser intensity necessary to create a constant number of ions passes through a minimum, as expected, for a three-photon energy equal to the unperturbed $6S\ensuremath{\rightarrow}6F$ transition energy. But for a higher ionization rate, involving larger laser intensities, a departure from expectation is observed which is interpreted as a consequence of an energy shift of the transition. The level shifts are demonstrated in another way by their strong effect on the experimental order of nonlinearity of the interaction, ${K}_{\mathrm{expt}}$. A sharp variation of ${K}_{\mathrm{expt}}$ is observed when the energy shift of the $6S\ensuremath{\rightarrow}6F$ transition exactly compensates the wavelength detuning of the laser. The latter method gives shift values which are found to be independent of the large variations of the interaction volume under conditions of resonance. Both methods give a linear dependence of the shifts on the laser intensity. Identification of these shifts with energy shifts of the $6S\ensuremath{\rightarrow}6F$ transition is subject to theoretical reservations and needs further experimentation.