Gorilla glue and spring clamps work wonderfully. The excess glue cleans up in water and sets in less than an hour. Premoisten the pieces being glued and it speeds up curing time. Best of all, it is gap filling and doesn't require solvent to clean up.
When you put the glue on/in use a small artists brush a #1 will work fine. Clamp the split sections together and allow to dry
The glue will in fact be stronger than the wood.
Any wood working glue will work with the same results as the Gorilla glue. I use Tite-bond.