The millimeter-wave band is a very attractive frequency band for the new generations of mobile cellular networks, i.e. 5G and 6G, due to its potential to support extremely high data rate transmissions. Innovative antenna solutions are needed to relieve the higher free space attenuation at these frequencies. Here, we propose a multibeam antenna based on a geodesic lens with stable aggregate gain characteristics in the far-field. An analytical model based on the physical path of the rays inside the lens is applied to achieve the field distribution in the aperture of the lens. This method is used to find the profile of a near-field focusing lens with a widened beam in the far-field. As a proof of concept, a seven-beam antenna has been designed. Due to the rotational symmetry of the geodesic lens, the antenna beams present similar characteristics over an extended sectorial coverage. The lens antenna has been manufactured, and its near-field focusing features were validated. The prototype has been assessed also in the far-field with a good agreement between model, simulations, and measurements. The main novelty of this prototype is to achieve a multiple beam coverage within +/- 67 degrees in the H-plane and +/- 20 degrees in the E-plane, with a gain roll-off smaller than 2 dB at 30 GHz and smaller than 3 dB at 35 GHz. These results validate the stable aggregate gain characteristics in the far-field of the proposed solution.