  The thorium nucleus with mass number $A=229$ has attracted much interest
because its extremely low lying first excited isomeric state at about $8$eV
opens the possibility for the development of a nuclear clock. However, neither
the exact energy of this nuclear isomer nor properties, such as nuclear
magnetic dipole and electric quadrupole moment are known to a high precision so
far. The latter can be determined by investigating the hyperfine structure of
thorium atoms or ions. Due to its electronic structure and the long lifetime of
the nuclear isomeric state, Th$^{2+}$ is especially suitable for such kind of
studies. In this letter we present a combined experimental and theoretical
investigation of the hyperfine structure of the $^{229}$Th$^{2+}$ ion in the
nuclear ground and isomeric state. A very good agreement between theory and
experiment is found for the nuclear ground state. Moreover, we use our
calculations to confirm the recently presented experimental value for the
nuclear magnetic dipole moment of the thorium nuclear isomer, which was in
contradiction to previous theoretical studies.
