My wired doorbell button broke so I bought a replacement (the actual
plastic button cracked and fell out) The doorbell worked fine before
the button broke.
I purchased a replacement button and removed the old one- here is what
stumps me:
As soon as the wires from the doorframe contact the screw terminals on
the new button, the doorbell chimes repeatedly. This happens at
contact- without the actual button being pressed.

As others have already mentioned, you probably bought a lighted button. The
filament in the bulb is conducting enough current to trigger your doorbell.

I encountered this same situation a few months ago at my in-laws. They have
a battery operated doorbell that wouldn't support a light anyway. So I just
cut the leads to the light on the button. It works fine now.

Anthony Watson
www.mountainsoftware.com
www.watsondiy.com