I have a 13" and a 10L. The 10L has a collet tube so I can describe what parts are needed but I have never adapted a longer tube to a shorter spindle although I have a longer tube and hope to do so one winter day.
It you have a threaded spindle, you will need a spindle taper to 5C taper adapter and a threaded collar to retain the adapter in the spindle. I forget the south bend names for these parts and do not have my parts manual handy. You will need to reduce the length of the over length tube. I would install the 5C adapter then insert the tube and a collet, thread the adapter onto the collet leaving enough threads to pull the collet tight then measure between the outboard end of the spindle and the tapered collar at the outboard end of the tube. This should tell you how much the tube needs to be shortened.
I have a tube from a 16" lathe and it has a bad thread where the collet threads into the tube. I plan to cut my tube to the necessary length and cut a new internal thread to engage the collet. The thread OD and pitch is available in the 5C collet standards. I remember someone posting them here a few years ago.
measurements from an actual 13" collet tube will help you but you should post the type spindle you have: threaded, D1-4 or L-0 as it will make a difference in the tube length.
