The joint at the corner, according to Gary Rogowski's The Complete Illustrated Guide to Joinery is a strengthened halved joint. It is fastened with nails Weber made himself. At the bottom of the sides, you can see a through tenon that secures the bottom of the chest to the sides. I don't know if the sides of the bottom are fitted into a groove.
A few more pictures by Gin are up on Picasa here. Thanks, Gin!
Check out Don Weber's website here.