It's never fun to have a bow break on you!!
I'm curious, the piece you were working on looked almost like a very narrow sapling; is that the case or are my eyes playing tricks on me? It might be easier to tiller a stave that comes from a wider diameter log/branch/sapling, if it's possible to get a hold of one.
Good luck either way and as old W. Churchill said, "Nevah, nevah, Evah Give up!"