When powered Ring Pro won’t stop chiming.
sounds like it's not wired correctly or it uses too much power. When the button is pressed, the line voltage drops, which is why the light on the button goes out. electrically, there are conductors and insulators, and a button switches between the two, whereas your ring is somewhere between a conductor and insulator (a load). Yopu old-fashioned bell is confused by this new electrical gray-area.

If your wiring is correct, and it's still "stuck", you have a few options.


a bigger doorbell transformer. Many are very weak, especally ones made after real buzzers and bells were phased out by chimes and speakers (80s?). Make sure the transformer has enough "watts", "VA", or "ma" (labels vary) to power the Ring. 
thicker doorbell wire: lower guage wire drops fewer volts over distance, so it might make up the difference to the ringer enough to avoid the low-voltage triggering state continuously.
disconnect the ringer and rely on your phone to notify you of visitors.