yup.
you could also make a huntworthy bow from balsa. It'll have 2foot wide limbs, but it might shoot.
you could even back it with bamboo, if you take a 1/16" wide piece.
but.
why back birch with bamboo if there are so much more, better suited backings around?
if it's a really bad-grained board, try maple as backing, and trap it additionally. As birch was usually used as backing wood, instead of compression wood, another well-suited wood would be ... indeed, birch.
As birch is so ridiciously weak in compression, you might even want to back it with an even lighter wood, ie. pine. Backing it with lighter woods will make it alot easier to make a high-performance birch bow.
I hope you have some experience with bamboo. I tried a bb-maple, and I just felt how the maple was crunched. the bow took 6" of set after being drawn a full 20".
personally, I wouldn't really give birch a try, exept as a backing on a compression-pine belly, like the saami bows, or as a trapped selfbow.
Nick