There is no adjustment on the other end, no. The end of the cable just attaches to a lever on the calipers. Does the handbrake work well on both wheels? Does sound like its coming up very high to be just a stretched cable. Do the two cables pull evenly? It may be that one cable has detached from the caliper or broken, and this would allow the lever to move higher.
You should be able to see the back of the calipers if you look round and behind the rear wheels (if your car's not too low), and see if the cables are attached ok (Im assuming your on disc brakes?).
Replacements are not expensive, about £10-15 each I think, and not too difficult to fit with basic tools/jack/axle stand