Legend,
You didn't specify your desired draw weight or I could have calculated that too, but since the thickness of the bow is determined by the Modulus of Rupture, the Modulus of Elasticity, and the radius of the fully bent limb as a result of its length and your draw length (28 inches?), I calculate that for American Elm of MOE = 1,340,000 and MOR = 11,800, the thickness should be 0.428 inches thick.
Now, bear in mind, this calculation is based upon published numbers (wooddatabase.com), so the numbers of the specific tree that your stave came from could vary somewhat. If you don't conduct a standard bend test on the sample that you wish to make a bow from, it's always wise to start a bit thicker than the calculated number and then work down to that during the tillering process, making sure to keep an eye on set as you go.
Keep in mind too, this calculation is good for a straight limb, pyramid design with no reflexed tips, but I cannot imagine how that would affect the thickness calculation assuming your bending radius did not change.
Cheers!