The isotope Th-229 is the only nucleus known to possess an excited state Th-229m in the energy range of a few electronvolts-a transition energy typical for electrons in the valence shell of atoms, but about four orders of magnitude lower than typical nuclear excitation energies. Of the many applications that have been proposed for this nuclear system, which is accessible by optical methods, the most promising is a highly precise nuclear clock that outperforms existing atomic timekeepers. Here we present the laser spectroscopic investigation of the hyperfine structure of the doubly charged Th-229m ion and the determination of the fundamental nuclear properties of the isomer, namely, its magnetic dipole and electric quadrupole moments, as well as its nuclear charge radius. Following the recent direct detection of this long-sought isomer, we provide detailed insight into its nuclear structure and present a method for its non-destructive optical detection.