Ammonia is a carbon-free fuel that can be produced from renewable energy sources and has the potential to replace fossil fuels, exerting a significant impact on the decarbonization of power production and propulsion industries. However, the challenge lies in the high NOx emissions, narrow flammability, and low flame speed of ammonia/air mixtures. In this paper, we study a novel concept of double-flame premixed co-combustion (DFPC) of ammonia and methane in a double-swirl premixed combustion burner, which results in low NOx emissions and high flame stabilization. Large eddy simulations using a detailed chemical kinetic mechanism and planar laser-induced fluorescence imaging of OH and exhaust gas NO emission measurements are employed to investigate the fundamental mechanisms behind flame/flame interactions and NO emissions. The main findings are: (a) NO emissions can be reduced by 90% using the DFPC concept along with a significant broadening of flammability; (b) the outer methane/air flame stabilizes the inner ammonia flame in the shear layer of the two flames; (c) combustion products and excess oxygen leaked across the shear layer decrease the equivalence ratio of the inner ammonia/air mixture, reducing the NO formation of close-to-stoichiometric ammonia/air flame but increasing the NO formation in the fuel-rich ammonia/air flames; (d) mixing of the combustion products from the inner and outer flames reduces the NO emissions in the flue exhaust gas.