Packing of graphs with small product of sizes

We show that for every ϵ>0, there exists n0=n0(ϵ) such that for every n>n0, two n-vertex graphs G1 and G2 with e(G1)e(G2)⩽(1−ϵ)n2 pack, unless they belong to a well-defined family of exceptions. This extends a well-known result by Sauer and Spencer.