The ionic conductivity and glass transition temperatures of nanostructured block copolymer electrolytes composed of polystyrene-b-poly(ethylene oxide) (SEO) doped with lithium bis(trifluoromethanesulfone)imide (LiTFSI) were studied in the small molecular weight limit (between 2.7 and 13.7 kg mol(-1)). In this range, the annealed conductivity exhibits a nonmonotonic dependence on molecular weight, decreasing with increasing molecular weight in the small molecular weight limit before increasing when molecular weight exceeds about 10 kg mol(-1). We show that annealed electrolyte conductivity is affected by two competing factors: the glass transition temperature of the insulating polystyrene (PS) block and the width of the conducting poly(ethylene oxide) (PEO) channel. In the low molecular weight limit, all ions are in contact with both PS and PEO segments. The intermixing between PS and PEO segments is restricted to an interfacial zone of width, lambda. Our experiments suggest that lambda is about 5 nm. The fraction of ions affected by the interfacial zone decreases as the conducting channel width increases. We also study the effect of thermal history on the conductivity of the block copolymer electrolytes. Our data suggest that long-range order impedes ion transport.