Using an ultracentrifuge rotor, the shift of the 14.4-keV Mössbauer absorption line of Fe57 in a rotating system was measured as a function of the angular velocity ω. An Fe57 absorber was placed at a radius of 9.3 cm from the axis of the rotor. A Co57 source was mounted on a piezoelectric transducer at the center of the rotor. By applying a triangularly varying voltage to the transducer, the source could be moved relative to the absorber. This arrangement makes possible the observation of the entire resonance line at various values of ω. The measured transverse Doppler shift agrees within an experimental error of 1.1% with the predictions of the theory of relativity. Possible sources of systematic errors are discussed.