Behind the cone are two very flexible wires, about 1 to 1.5" long.
These are soldered on to a tag mounted on the woofer basket (the metal frame behind the cone).
It sounds like one of those solder joints has failed.
Sometimes, where the wire joins the voice coil fails too.
But I suspect it's the first option - if you remove the woofer (carefully!!!) you shuold be able to see straight away.
Tony.