Preheat the oven to 350 degrees. Grease a 9 x 13 in pan.

In a large bowl combine sugars, flour, and salt and whisk together.

In a medium sized sauce pan combine butter, root beer, and cocoa. Bring to a boil and then pour over sugar and flour, whisking thoroughly until no lumps remain. Stir in buttermilk, baking soda, and beaten eggs and stir gently until smooth. Pour mixture into the greased pan and bake for 30 minutes or until toothpick inserted in the center comes out clean.

For the frosting, combine root beer, butter, and cocoa in a medium saucepan. Bring to a boil and then whisk in powdered sugar, 1 cup at a time, until mixture is thickened but still pourable (you want it to be like a glaze). Pour over cake while still warm, saving some for drizzling over floats.

To assemble floats, place bite size pieces of cake (frosted or unfrosted) at the bottom of a tall glass. Layer with a scoop of ice cream, then more cake bites, then more ice cream until the glass is full. Pour root beer over all and drizzle with extra frosting.