A new method is presented for inverting the integrated phase-shift data obtained from an axisymmetric refractive index field. The method is of the onion-peeling type, and was developed specifically to accommodate the features of fringe patterns obtained from reference-beam interferometry. The method determines the refractive index as a function of radius over a series of irregularly spaced, concentric rings whose edges correspond to the fringe location. The refractive index is presumed to vary in a linear fashion in all rings except the central ring, over which a hyperbolic distribution is presumed. The accuracy of the method is examined by the inversion of a fringe pattern obtained from an analytical function of refractive index. It is shown to give very accurate results for as few as four data points, corresponding to four fringe crossings in the interferogram. This method appears to be superior to other considerably more complex methods for the inversion of axisymmetric interferograms, and it requires no smoothing of the original data.