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We present the design of a flat lens, made by a conventional material and an epsilon near-zero metamaterial, to plug up the aperture of a short horn antenna, in order to achieve radiation performances similar to the ones of the corresponding optimum horn over a broad frequency range. Lens operation is based on the phase-compensation concept: phase-fronts of the field propagating along the short flare of the horn propagate with different phase velocities in the two lens materials, resulting in an uniform phase distribution on the aperture. Starting from the theoretical study of the transmission properties of a bulk epsilon near-zero slab, we derive the analytical formulas for the design of the flat lens and validate them through full-wave numerical simulations. Then, a realistic version of the lens, realized with a wire-medium and exhibiting a near-zero real part of the effective permittivity in the frequency range of interest, is presented. Considering two examples working in the C-band, we show that the lens can be designed for both conical and pyramidal horn antennas. In both cases, the length of the horns is half the one of the corresponding optimum versions, while the obtained radiation performances are similar to those of the optimum horns over a broad frequency band. This result may open the door to several interesting applications in satellite and radar systems.