A solvent-free solid-polymer electrolyte based on a cross-linked poly(ethylene oxide-propylene oxide) random copolymer doped with LiN(SO2CF3)(2), was studied using multinuclear NMR and ionic conductivity. The NMR spin-lattice relaxation times, T-1, of the bulk polymer (H-1), the lithium ion (Li-7) and the anion (F-19) were analyzed using single exponential analysis above the glass transition temperatures. Since the temperature dependent H-1 and Li-7 NMR T-1 values had minima, the reorientational correlation times were obtained for the segmental motion of the CH2CH2O/CH2CH(CH3)O moiety of the bulk polymer and the hopping motion of the lithium ions correlated with the segmental motion. The spin-spin relaxation of the anion signals appeared single exponential with respect to time, whereas that of the polymer and the lithium echo signals were at least bi-exponential. Since both the spin-lattice and spin-spin relaxation of the anion indicated a single component, the self-diffusion coefficients, D, were measured using F-19 pulsed-gradient spin-echo (PGSE) NMR measurements. Although the PGSE attenuation data appeared single exponential at each value of the separation between the gradient pulses, Delta, the measured D values had a Delta-dependence. Phenomenologically, the anion diffuses quicker in a shorter range and the activation energy of the shorter-time diffusion is smaller than that of the longer-time diffusion. The apparent self-diffusion coefficients became smaller for longer Delta and approached a constant when Delta was longer than 0.05 s. The mean square displacements of the anion were inconsistent with standard diffusion models including "anomalous diffusion" as found for a neutral particle diffusing in a fractal network [i.e., < r(2)(Delta)>proportional to Delta(kappa) with kappa < 1(kappa equivalent to 2/d(w) where d(w) is the random walk fractal dimension)]. The apparent diffusion coefficients of the lithium ions at Delta=0.02 s are almost independent of temperature and smaller than the corresponding diffusion coefficients of the anion. Since the activation energy of the anion determined for Delta longer than 0.05 s correlates well with those obtained from the ionic conductivity, the ion conduction in the solid-polymer medium is driven mainly by fast transfer of the anions.