Doorbell Wiring Problem
Most likely, the doorbell button has failed.  

There will always be voltage across the doorbell button unless it is pushed.  That's because the circuit would be complete, but for the button.  I would expect to see 18 volts (or bare transformer voltage), but with an electronic chime, I could see that value being a little funny.  

You can test this out; short the 2 wires to the doorbell, that is the same as ringing the doorbell.