Standing shocks are believed to be responsible for stationary Type II solar radio bursts, whereas drifting Type II bursts are excited by moving shocks often related to coronal mass ejections (CMEs). Observations of either stationary or drifting Type II bursts are common, but a transition between the two states has not yet been reported. Here, we present a Type II burst which shows a clear, continuous transition from a stationary to a drifting state, the first observation of its kind. Moreover, band splitting is observed in the stationary parts of the burst, as well as intriguing negative and positive frequency-drift fine structures within the stationary emissions. The relation of the radio emissions to an observed jet and a narrow CME was investigated across multiple wavelengths, and the mechanisms leading to the transitioning Type II burst were determined. We find that a jet eruption generates a streamer-puff CME and that the interplay between the CME-driven shock and the streamer is likely to be responsible for the observed radio emissions.