The results of C-13 nuclear magnetic resonance (NMR) measurements on alkali fullerides K(x)C60 are reported. The NMR spectra demonstrate that material with 0 < x < 3 is in fact a two-phase system at equilibrium, with x = 0 and x = 3. NMR lineshapes indicate that C60(3-) ions rotate rapidly in the K3C60 phase at 300 K, while C60(6-) ions in the insulating K6C60 phase are static on the time scale of the lineshape measurement. The temperature dependence of the C-13 spin-lattice relaxation rate in the normal state of K3C60 is found to be characteristic of a metal, indicating the important role of the C60(3-) ions in the conductivity. From the relaxation measurements, an estimate of the density of electronic states at the Fermi level is derived.