  We present precision measurements of the hyperfine intervals in the 5D3/2
manifold of a single trapped Barium ion, 137 Ba+ . Measurements of the
hyperfine intervals are made between mF = 0 sublevels over a range of magnetic
fields allowing us to interpolate to the zero field values with an accuracy
below a few Hz, an improvement on previous measurements by three orders of
magnitude. Our results, in conjunction with theoretical calculations, provide a
30-fold reduction in the uncertainty of the magnetic dipole (A) and electric
quadrupole (B) hyperfine constants. In addition, we obtain the magnetic
octupole constant (C) with an accuracy below 0.1 Hz. This gives a subsequent
determination of the nuclear magnetic octupole moment, {\Omega}, with an
uncertainty of 1% limited almost completely by the accuracy of theoretical
calculations. This constitutes the first observation of the octupole moment in
137 Ba+ and the most accurately determined octupole moment to date.
