Summary: An efficient and robust algorithm and a Matlab code ratdisk are presented for rational interpolation or linearized least-squares approximation of a function based on its values at points equally spaced on a circle. The use of the singular value decomposition enables the detection and elimination of spurious poles or Froissart doublets that commonly complicate such fits without contributing to the quality of the approximation. As an application, the algorithm leads to a method for the stable computation of certain radial basis function interpolants in the difficult case of smoothness parameter \(\epsilon\) close to zero.