if you don't mind me asking, how is it broken? did the cord fray, did it actually disconnect, or you can't visually tell the problem but it just no longer re-charges the phone? Also, are you charging directly from an outlet or from a computer? From a USB port on a computer? What kind of computer?
The reason I ask is because I have a similar problem. I purchased my iphone 5 about a month ago, and I've already gone through 2 usb charge/sync cables. It is absolutely ridiculous, the cable has to be in a "particular" position in order to charge/sync when it is attached to my Macbook Air - furthermore, there's evidence of a slight burn/black mark on the actual pins on the end going into the iphone. The 2nd cord that charges the phone in my room directly from the wall has no problems.
When I spoke to the individual at the Apple Genius Bar, he said I had to bring in everything (macbook air, iphone, cords etc.) before he'll look at the problem. However, after combing through this community, it seems like this is a known issue.