Wrap the box in butcher paper. There are two reasons to wrap it in butcher paper. First, it will give you a solid color palette underneath the fabric; secondly, the glue and fabric will stick better to the butcher paper than to the cardboard. Wrap the box as you would a gift package and use scotch tape to secure. Then wrap the box in the burlap or muslin fabric. Wrap it as you would a gift package and use hot glue to secure.
Measure from the top center of the box, down to the middle of each side to the bottom of the box. Cut the birch bark into four rectangular strips to those two sizes (one for the long side and one for the short side). Make it so all four pieces slightly overlap in the center of the box. Don't cover the bottom of the box, just bring the bark down the sides.
If you can, the night before you start this project, make the loops for the bow. Cut two pieces of bark the same size. Curl one end of each piece onto itself and hold it with a paperclip. If you let the loops sit overnight, they will hold their shape better.