We measure the hyperfine splitting of the 9S(1/2) level of Fr-210, and find a magnetic dipole hyperfine constant A=622.25(36) MHz. The theoretical value, obtained using the relativistic all-order method from the electronic wave function at the nucleus, allows us to extract a nuclear magnetic moment of 4.38(5)mu(N) for this isotope, which represents a factor of 2 improvement in precision over previous measurements. The same method can be applied to other rare isotopes and elements.