........ There's a break in wiring and hiring an electrician would really be a high expense. 
Bad wire that's a bummer. I take it you where able to maybe check the Bell transformer to see if it died.
If its the wire... the wire itself is likely just two loosely twisted solid wires. Often the wires were referred to as "phone wire" around that same era. The wire
should feed through the attic, basement, crawl space, or wall to the chime unit. Since it is an older unit...
a transformer could be mounted on a utility box somewhere along the way. But the location of the bell/chime unit (often in a closet, nearby wall, or centrally located wall) should make the wire route somewhat obvious.
This is low voltage wiring and...
if its the wiring it can be repaired (or replaced) for little or nothing. Often the damage is caused when phone wire repair/upgrades are done... because the doorbell wire is mistaken for phones. Sometimes a broken wire can be found in a basement laundry area... thanks to coat hangers.
I think
Brian H is likely correct about the transformer (or bell/chime unit).
But ether way... this sounds like a fun repair.
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Topic: How does X10's Wireless Doorbell work (Read 19803 times)