An analysis of the cylindrical brush of an isolated comb copolymer molecule, consisting of a semiflexible backbone and rodlike side chains, is presented. Using a mean-field approach and a simplyfying assumption, which is tested by computer simulations, we find that the persistence length of the brush, lambda, scales as lambda proportional to L-2/ln L for large values of the side chain length L. In the cylindrical brush regime the order parameter of the rods is negative, implying that the rods orient normal to the cylindrical axis.