The Doppler effect, the shift in the frequency of sound due to motion, is present in both classical gases and quantum superfluids. Here, we perform an in situ, minimally destructive measurement, of the persistent current in a ring-shaped, superfluid Bose-Einstein condensate using the Doppler effect. Phonon modes generated in this condensate have their frequencies Doppler shifted by a persistent current. This frequency shift will cause a standing-wave phonon mode to be dragged along with the persistent current. By measuring this precession, one can extract the background flow velocity. This technique will find utility in experiments where the winding number is important, such as in emerging atomtronic devices.