The origin of homochirality in living systems is often attributed to the generation of enantiomeric differences in a pool of chiral prebiotic molecules, but none of the possible physiochemical processes considered can produce the significant imbalance required if homochiral biopolymers are to result from simple coupling of suitable precursor molecules. This implies a central role either for additional processes that can selectively amplify an initially minute enantiomeric difference in the starting material, or for a nonenzymatic process by which biopolymers undergo chiroselective molecular replication. Given that molecular self-replication and the capacity for selection are necessary conditions for the emergence of life, chiroselective replication of biopolymers seems a particularly attractive process for explaining homochirality in nature. Here we report that a 32-residue peptide replicator, designed according to our earlier principles, is capable of efficiently amplifying homochiral products from a racemic mixture of peptide fragments through a chiroselective autocatalytic cycle. The chiroselective amplification process discriminates between structures possessing even single stereochemical mutations within otherwise homochiral sequences. Moreover, the system exhibits a dynamic stereochemical 'editing' function; in contrast to the previously observed error correction, it makes use of heterochiral sequences that arise through uncatalysed background reactions to catalyse the production of the homochiral product. These results support the idea that self-replicating polypeptides could have played a key role in the origin of homochirality on Earth.