  We perform precision measurements on the 5D$_{5/2}$ manifold hyperfine
intervals of a single trapped ion, $^{137}$Ba$^+$. RF spectroscopy is used to
measure the hyperfine intervals to an accuracy of a few Hz. Our results provide
a three orders of magnitude improvement in accuracy over previous work and also
provide a 10-fold improvement in the value of $g_J$ for this level. These
results complement our previous work on the 5D$_{3/2}$ manifold of
$^{137}$Ba$^+$, providing an independent measurement of the nuclear octupole,
and a consistency check on atomic structure calculations.
