I have a DC01 (circa 1998) and experienced the cleaner cutting out, every now and then. To the end where it burnt a hole in the cable at the insertion point below the on/off switch. Take away the drum/dust collector part and there is a screw of the six star variety which holds a plate which when removed reveals the cable connection and everything else. (Try in your drill/screw bit collection for the six star shaped head.) Cut the cable, removing the burnt frayed part and rewired. The brown cable goes into a crimper in a plastic sheath. Slip a small flat screwdriver inside this carefully and you should be able to remove the crimper.
replace the crimper, as some folks have said with replacements from Maplins or try and prize open the crimped parts and replace with new end of cable wire.
Re-assemble. No need to remove the on/off switch for this DC01.
Peter Nodsy, November 2005