We study a two dimensional system whose two-point correlation function is zero for any non-zero distance, as for a perfectly random sequence of coin flips. Consequently, its diffraction pattern has no singularities of any kind, and so would be considered disordered. By mapping this onto a spin model we have constructed a short-range Hamiltonian for which this is the ground state. To our knowledge, this is the first Hamiltonian with deterministic, nearest-neighbor interactions whose ground state is disordered in this sense.