  We identify a potential means to extract the $^{229g}$Th $\rightarrow$
$^{229m}$Th nuclear excitation energy from precision microwave spectroscopy of
the $5F_{5/2,7/2}$ hyperfine manifolds in the ion $^{229g}$Th$^{3+}$. The
hyperfine interaction mixes this ground fine structure doublet with states of
the nuclear isomer, introducing small but observable shifts to the hyperfine
sub-levels. We demonstrate how accurate atomic structure calculations may be
combined with measurement of the hyperfine intervals to quantify the effects of
this mixing. Further knowledge of the magnetic dipole decay rate of the isomer,
as recently reported, allows an indirect determination of the nuclear
excitation energy.
