The relationship between ionic conductivity, morphology, and rheological properties of polystyrene-block-poly(ethylene oxide) copolymers (SEO) doped with a lithium salt, Li[N(SO₂CF₃)₂], is elucidated. We focus on lamellar samples with poly(ethylene oxide) (PEO) volume fractions, ϕ, ranging from 0.38 to 0.55, and PEO block molecular weights, M_PEO, ranging from 16 to 98 kg/mol. The low-frequency storage modulus (G‘) at 90 °C increases with increasing M_PEO from about 4 × 10⁵ to 5 × 10⁷ Pa. Surprisingly, the conductivity of the SEO/salt mixtures with the molar ratio of Li to ethylene oxide moieties of 0.02 σ, also increases with increasing M_PEO, from 6.2 × 10^-5 to 3.6 × 10^-4 S/cm at 90 °C. We compare σ with the conductivity of pure PEO/salt mixtures, σ_PEO, and find that σ/[ϕσ_PEO] of our highest molecular weight sample is close to 0.67, the theoretical upper limit for transport through randomly oriented lamellar grains.