Lithium-sulfur batteries are attractive due to their high theoretical specific energy, but the dissolution of lithium polysulfide intermediate species formed during discharge results in capacity fade and limited cycle life. In this study we present the first measurements of ionic conductivity of the polysulfides in a nanostructured block copolymer. The morphology, thermal properties, and the conductivities of polystyrene-b-poly(ethylene oxide) (SEO) containing lithium polysulfides, Li2Sx (x = 4, 8), were studied using small-angle X-ray scattering (SAXS), differential scanning calorimetry (DSC), and ac impedance spectroscopy. We also measured conductivities of mixtures of poly(ethylene oxide) (PEO) and Li2Sx. X-ray absorption spectroscopy was used to confirm the nature of dissolved polysulfides. SAXS measurements on SEO/Li2Sx mixtures indicated that all samples had a lamellar morphology. DSC measurements indicated that SEO/Li2S8 interactions were more favorable than SEO/Li2S4 interactions. The effect of nanostructure on transport of Li2Sx was quantified by calculating a normalized conductivity, which is proportional to the ratio of the conductivity of SEO/Li2Sx to that of the PEO/Li2Sx. The normalized conductivities of both polysulfides peaked at intermediate concentrations. The efficacy of block copolymer electrolytes in Li-S batteries was evaluated by comparing ionic conductivities of polymer electrolytes containing Li2Sx with those containing lithium bis(trifluoromethanesulfonyl)imide (LiTFSI), a common salt used in PEO-based battery electrolytes. The transport of Li2Sx species in SEO is suppressed by factors ranging from 0.4 to 0.04 relative to LiTFSI, depending on x and salt concentration. To our knowledge, this study represents the first systematic investigation of the effect of molecular structure of polymer electrolytes on polysulfide migration.