Single-walled carbon nanotube diodes are fabricated in a split-gate geometry with electron ͑n͒ and hole ͑p͒ regions separated by a central region. With the central region gated p or n type the diodes "leak" at low voltages, likely due to tunneling across the smaller depletion region. With the central region intrinsic, nearly ideal diode behavior is observed. Comparison to theory for a one-dimensional diode yields the band gap of the tube and the transmission coefficient through the junction. In reverse bias, the breakdown voltage depends weakly on temperature and nanotube diameter. Comparisons are made to predictions for Zener tunneling and avalanche breakdown.