Transport of ions in polymer electrolytes is of significant practical interest, however, differences in the transport of anions and cations have not been comprehensively addressed. We present measurements of the electrochemical transport properties of lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) in poly(ethylene oxide) (PEO) over a wide range of PEO molecular weights and salt concentrations. Individual self-diffusion coefficients of the Li+ and TFSI- ions, D+ and D-, were measured using pulsed-field gradient nuclear magnetic resonance both in the dilute limit and at high salt concentrations. Conductivities calculated from the measured D+ and D- values based on the Nernst-Einstein equation were in agreement with experimental measurements reported in the literature, indicating that the salt is fully dissociated in these PEO/LiTFSI mixtures. This enables determination of the molecular weight dependence of the cation transference number in both dilute and concentrated solutions. We introduce a new parameter, s, the number of lithium ions per polymer chain, that allows us to account for both the effect of salt concentration and molecular weight on cation and anion diffusion. Expressing cation and anion diffusion coefficients as functions of s results in a collapse of D+ and D- onto a single master curve.