Hi! I did the same but using a small DC relay. Instead of the bell, I have a full bridge rectifier with a electrolytic capacitor on the DC side and the relay. This way the Ring is powered and when send the bell signal the relay closes and activate my electronic doorbell.
To answer Prembo, the coil is not energized since the Ring does not let enough current on the line unless it wants to ring the bell.
Below a crude sketch of how it works. I used the DC relay since was a lot smaller and everything fitted in a 1 inch cube.
