With some reservations, I think there might be a possibility that this could be done safely. I'd be leery of doing this if the cut/weld area was anywhere between the breech and several inches ahead of the forearm area where your fingers could be damaged if the repair failed. My first several shots would definitely be done with heavy loads and the gun tied to a tire and a long string to pull the trigger... you know, American Proof House. Then if it held, low pressure loads only. If the cut was anywhere near the ribs, the heat would almost certainly melt the solder joint. If it was on the side of the barrel away from the ribs, one could do the repair with the good tube and the ribs just under the surface of a tank of water so that the heat could never separate the ribs. I would want to have a close fitting plug of copper bar inside the barrel that would act as both a heat sink, and provide a back-up surface that the weld would not stick to. This is commonly done when TIG welding thin steel. The purpose is not to keep the thoriated tungsten from sticking because you absolutely never want the tip of the tungsten to touch anything or it will become contaminated and need reground. I might also consider filling the barrel with Argon-CO2 gas to prevent scale formation.
After welding and cooling, the copper plug would be driven out and the repair area lightly honed inside and dressed outside. Reblueing would be necessary of course, and it wouldn't be a shock to find that the weld took blueing differently than the base metal.
I said I had some reservations about this, safety being foremost. I don't think barrel steel would be an air hardening variety that would become brittle from welding and cooling, but I couldn't say that for certain. Before anyone pillories me for thinking this could work, I'd ask them to consider that Damascus barrels are composed of a lot more welds than filling in 3/8" of hacksaw kerf.
