We prove the following formula for the ground state energy density of a dilute Bose gas with density ρ in 2 dimensions in the thermodynamic limit (Formula presented.) as ρa²→0. Here Y=|log(ρa²)|^-1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.