Solutions for frequency-invariant beamforming with concentric circular arrays are used in many applications, including microphone arrays and audio communication. However, existing methods focus on the azimuth-beamwidth and consider uniformly spaced circular arrays. This paper presents an approach to designing a constant elevation-beamwidth beamformer for nonuniform concentric ring arrays. The suggested beamformer aims to maximize the array directivity factor while maintaining fixed beamwidth over a wide range of frequencies. The introduced methodology simultaneously selects the ring placements and designs the beamformer weights that achieve optimal performance. We present the problem constraints and cost function, resulting in a sparse beamformer design. In addition, a uniformly spaced beamformer is incorporated into the optimization cost function to attain improved performance. Time-domain implementation of the ideal beamformer is also presented in this work. Furthermore, in the proposed array configuration, all sensors on each ring share the same weight value. Thus, the computational complexity and the physical hardware required in a physical setup are significantly reduced. Experimental results demonstrate the advantages of the nonuniform beamformer compared to a uniform beamformer in terms of directivity index, white noise gain, and sidelobe attenuation.