We have performed precision measurements of the hyperfine intervals of the 5D3/2 and 5D5/2 manifolds of a single trapped ion, 137Ba+. We have used rf spectroscopy to measure the hyperfine intervals to an accuracy of a few Hz. Our results provide a three orders of magnitude increase in accuracy over previous work. These results allow the determination of the nuclear octupole without using any theoretical correction factor caused by the mixing between 5D5/2 and 5D3/2 states. We have measured the nuclear magnetic octupole moment down to an accuracy of 1\%.