I would try wood glue and clamp it good until it dries. Use a wet Q-Tip to clean up the glue that squeezes into the threads and bowl area and a wet cloth to clean up the outside all before it dries. Then let it sit for 24 or 48 hrs while it dries. Should be good to go after that. Wood glue is strong stuff as long as you using it on clean wood.
If you are still concerned about it you could make some tooth pic dowels and install them before you glue it all together but I doubt that would be necessary.