Originally Posted by
nashvillebill
I would just have the new section welded with a double V butt weld at the joints, meaning the old section is chamfered at the end, and the new section is chamfered, then the weld fills up those areas. The resulting weld in theory can be just as strong as the parent metal. No internal sleeves.
The point of creating a groove (you mentioned welding a double V groove) is that it lets you penetrate into thick material when welding. In this case the tube is very thin so penetration wouldn't be an issue. In fact you would want the tube thicker at the weld so as not to get "melt-through" at the joint (hence the sleeve suggestion). A sleeve would act as reinforcement, a heat sink, and would keep the tubes square to each other during fitment.
Also, a mod might want to change the MIG mention in the title to TIG. As far as I know MIG is not used for bike fabrication.