This article reports the synthesis, spontaneous self-assembly, highly efficient drug encapsulation, and glutathione (GSH)-triggered intracellular sustained drug delivery of an ABA-type amphiphilic triblock copolymer, namely, polyglycerol-b-poly(disulfide)-b-polyglycerol (PG-b-PDS-b-PG). The bioreducible PDS block with reactive pyridyldisulfide groups present at the chain terminals was attached to thiol-terminated heterotelechelic PG by a thiol–disulfide exchange reaction producing the amphiphilic PG-b-PDS-b-PG. It formed a stable polymersome in aqueous medium with a critical aggregation concentration of 0.02 mg/mL and average hydrodynamic diameter (Dh) of 230 nm and showed highly efficient and stable encapsulation of doxorubicin (Dox) with a remarkably high drug loading efficiency (DLE) and drug loading content (DLC) of 54% and 16%, respectively. Fluorescence spectroscopy studies revealed GSH-triggered drug release and strong dependence of the release kinetics on the GSH concentration due to degradation of the amphiphilic block copolymer and disassembly of the polymersome. MTT assay indicated excellent biocompatibility of the block copolymer as >90% cells (HeLa or MDA-MB-231) were found to be alive after 96 h of incubation with a polymer concentration of up to 1.0 mg/mL, which was further validated by the hemolysis assay. Cytotoxicity assay of the Dox-loaded polymersome exhibited time and dose-dependent sustained killing of HeLa as well as MDA-MB-231 cells wherein after 48 h of incubation >50% cell killing was noticed with a Dox concentration of ∼4.0 and ∼8.7 μg/mL, respectively, while the free Dox showed faster cell killing. Flow cytometry and live cell fluorescence microscopy studies revealed time-dependent cellular uptake of the drug-loaded polymersome followed by diffusion of the drug to the nucleus. Cells with artificially enhanced GSH were killed at a much faster rate indicating that intracellular GSH-triggered disassembly is the key drug release mechanism.