We present a Bessel beam launcher based on a radial leaky-wave antenna operating at millimeter waves. With respect to a design method proposed by one of the Authors in a lower frequency range, a new approach is chosen here, as an alternative to a simple scaling of all the geometrical parameters. In order to avoid inacceptable levels of losses and possible fabrication issues due to an extremely thin substrate, our design is based on the suitable excitation of a higher-order leaky mode. Closed-form equations are derived in order to simplify the design, providing reliable values for the physical parameters once the near-field characteristics are assigned. Simulated results are shown to validate the proposed procedure.