I've done this a few times on a GT sts and a Klein.
My technique involves a cost hanger. Unwind the hook and straighten it all out. Bend it to shape that will allow you to push it through one end of the frame, and patience to get it to come through the other side.
The cost hanger is almost the perfect diameter compared to a cable, so once you have it running through the frame, you can push a length of the sheath down the hanger then pull it through (,well push). Once you have that through, remove the hanger and slide your actual cable through the sheath. Job done. It's much harder a job to do this on a GT sts than a Klein, I've done both this way.