An annular periodic leaky-wave antenna (LWA) fed by a simple azimuth-symmetric source is designed to generate a high-gain omnidirectional conical-beam pattern, which scans with frequency over a wide angular range. The proposed structure is defined by a finite metallic radial strip grating printed on a grounded dielectric slab, which supports an n = 0 cylindrical leaky wave (CLW). The distinctive features of CLWs supported by such a truncated structure are also highlighted and discussed. The directional far-field pattern generated by the proposed LWA, in conjunction with its nondiffracting and wideband behavior in the near field (as previously reported by the authors), defines an original dual-operational LWA. Possible applications include next-generation wireless power transfer systems that provide functionality in both the near and far field, vehicle roof-mounted antennas for basestation data connectivity as well as future short-range near-field communications, and object tracking by ceiling-mounting devices for indoor localization.