In this contribution, amphiphilic star copolymers (H40-star-PCL-a-PEG) with an H40 hyperbranched polyester core and poly(ε-caprolactone)-a-poly(ethylene glycol) copolymer arms linked with acetal groups are synthesized using ring-opening polymerization and a copper (I)-catalyzed alkyne-azide cycloaddition click reaction. The acid-cleavable acetal groups between the hydrophilic and hydrophobic segments of the arms endow the amphiphilic star copolymers with pH responsiveness. In aqueous solution, unimolecular micelles can be formed with good stability and a unique acid degradability, as is desirable for anticancer drug carriers. For the model drug of doxorubicin, the in vitro release behavior, intracellular release, and inhibition of proliferation of HeLa cells show that the acid-cleavable unimolecular micelles with anticancer activity can be dissociated in an acidic environment and efficiently internalized by HeLa cells. Due to the acid-cleavable and biodegradable nature, unimolecular micelles from amphiphilic star copolymers are promising for applications in intracellular drug delivery for cancer chemotherapy.