The region of existence of neon clathrate hydrates is an actual problem of hydrate chemistry. The current work presents theoretical study of the equilibrium formation conditions of pure neon clathrate hydrates and double clathrate hydrates of neon-methane mixture. The structures and properties of double clathrate hydrates were described within the scope of the previously developed molecular clathrate hydrate model that takes into account the influence of guest molecules on the host lattice, interaction of guest molecules between themselves, and the possibility of multiple filling of host lattice cages by guest molecules. The model makes it possible to find an equilibrium state and thermodynamic properties of clathrate hydrates at given values of p and T. In the present work, we considered the properties of double clathrate hydrates in the range of pressures from 0 to 4 kbar at 250 K. The results of modeling have shown that the mass fraction of neon in double clathrate hydrate of Ne and CH4 mixture of cubic structure I (sI) can reach 26%, and 22.5% in double hydrate of cubic structure II (sII) even at a low methane concentration (1%) in gas phase, at high pressure. It is shown that in double clathrate hydrates of the Ne and CH4 mixture at high pressures, phase transition sII-sI can occur.