FIrst you need to determine where the break or frayed area is. Is it the plug end? With headphones on and music playing, wiggle the plug and see if a channel goes in and out. That should be a simple thing to replace -- you could probably DIY with simple soldering skills and the right replacement plug, if you're not too concerned with visual aesthetics.
Here's a new thread about replacing the plug.
http://www.head-fi.org/forum/thread/522864/recabled-phones-can-balance-be-affected
Often on AKG cans it's the cup end. My old K240DF cable went out (channel loss) so I sent it to AKG/Harman to replace -- $70 incl post. Probably should have sold them instead, recabling cost more than the cans later brought me at Ebay. Prices for recabling seem to vary considerably. Getting the housing enclosure off is the first problem, then you have to be able to detach old cable and resolder new one without damaging the headphone, so this is not as easy as the plug..
As for who besides Customer Service does stock or simple recables (as opposed to balanced or replacement by much more expensive cable), I dunno. I'd love to see a list. Try searching on "recabling service" and the like. zombie_x = http://www.zombie-x.com/zxamateurcablesrev2/ = is active on the board and prices look very reasonable. He owns AKG cans including one of the K240 models himself so is likely familiar with yours.
Edited by renlute - 11/17/10 at 6:43pm