Make sure that you are working in a risk free environment by turning off power at the main.
Remove the screws holding the doorbell button in place with a screwdriver. You may have to pry loose a plastic cover hiding the screws.
Use a wire cutter to disconnect one of the wires attached to the button. Since polarity does not count when dealing with a doorbell, you can cut any of the two wires.
Turn the power back on and check if the doorbell still rings. If it has stopped ringing, the problem was being created by the doorbell button. Replace the button and the doorbell would start working normally.
If that doorbell is still ringing after you have cut one of the wires connected to the button, then the wires are probably the cause of the problem. Turn off the power once again and look for frayed insulation that might be causing the wires to make contact with each other.
If you find such a short, fix it by insulating the bare part of frayed wires. Use a piece of electrician’s tape for this purpose. If you are unable to locate a short, replace the wires with new ones. Turn the power back on and the doorbell should work normally.