We determine the magnetic dipole mu = 0.360(7)mu(N) and the electric quadrupole Q = 3.11(6)eb moments of the Th-229 nucleus by combining our high-precision calculations of the hyperfine constants with measurements reported in Campbell et al. [Phys. Rev. Lett. 106, 223001 (2011)]. We find that the previous value mu = 0.46(4)mu(N) [Gerstenkorn et al., J. Phys. (Paris) 35, 483 (1974)] is incorrect by 25%. We report a method for determining the accuracy of theoretical hyperfine constants B/Q and demonstrate that it can be used to extract the electric quadrupole moment Q with a 1%-2% uncertainty for a large number of nuclei. This approach allowed us to identify 40% inconsistencies in measurements of Ra+ hyperfine constants B.