Analytical expressions for the resonant frequencies in an electromagnetic cylindrical cavity with an internal off-axis electrically small sphere are derived, for both magnetic and electric modes. The walls of the cylindrical cavity and the small sphere are perfectly conducting. Cylindrical vector wave functions and related translational addition theorems, as well as expansion of cylindrical wave functions in terms of spherical ones, are used. The results are useful in problems connected with excitation or probing of resonant cavities. Some remarks are made about the best positioning of the probe, for more exact measurement of the resonant frequency of any mode in the empty cavity. Graphical results for some of the lower order modes are given, for various values of the parameters and for both kinds of modes.