Cheap way to make cardboard box waterproof?
How about plastic bags?   You will need 2.  Put inner box inside one.  Leave bag open at top so concrete can push air out of the bag and flatten it against inner box.  

Put bagged box inside another bag.  Then double bagged box inside outer box.  

Pour concrete between the 2 bags.  Bags will flatten against boxes as they fill and contain concrete between the 2 boxes.  Peel off bags after concrete sets.  You may get some aesthetically pleasing wrinkles in the final product.

Added bonus: it will be easy to retrieve boxes from final product.

@manassehkatz is right about the concrete collapsing the boxes.  You could address this with a hole (or a pile of sand).  Dig a hole in the ground large enough to contain planter.  Fill inner box with dirt to fortify it against collapse.  Put box/bag mold into the hole and put dirt down around outer box.  That will keep it from bursting outwards from weight of wet concrete.  Or you could just fill inside box with sand and wrap outer box with ropes.