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We describe the development of two circularly symmetric antennas with high polarization purity and low spill-over. Both were designed to be used in an all-sky polarization and intensity survey at 5 GHz (the C-Band All-Sky Survey, C-BASS). The survey requirements call for very low cross-polar signal levels and far-out sidelobes. Two different existing antennas, with 6.1-m and 7.6-m diameter primaries, were adapted by replacing the feed and secondary optics, resulting in identical beam performances of 0.73° FWHM, cross-polarization better than - 50 dB, and far-out sidelobes below -70 dB. The polarization purity was realized by using a symmetric low-loss dielectric foam support structure for the secondary mirror, avoiding the need for secondary support struts. Ground spill-over was largely reduced by using absorbing baffles around the primary and secondary mirrors, and by the use of a low-sidelobe profiled corrugated feedhorn. The 6.1-m antenna and receiver have been completed and tested. Results show that the co-polar beam matches the design simulations very closely in the main beam and down to levels of - 80 dB in the backlobes. With the absorbing baffles in place the far-out ( >; 100°) sidelobe response is reduced below -90 dB. Cross-polar response could only be measured down to a noise floor of - 20 dB but is also consistent with the design simulations. Temperature loading and groundspill due to the secondary support were measured at less than 1 K.