We show that a (weighted) Carleson embedding from the bi-torus to the bi-disc is equivalent to a simple “box” condition, for product weights on the bi- disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale includes the usual Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the “box” condition does not suffice for the embedding to hold. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.