The formation of collimated outflows or jets in planetary nebulae (PNe) is not well understood. There is no evidence for active accretion discs in PNe, making it difficult to decide which of the several proposed jet formation scenarios may be correct. A handful of wide binary central stars of PNe are known to have accreted carbon and slow neutron capture (s-process) enhanced material, the immediate progenitors of barium stars; however, no close binary analogues are known to have passed through a common-envelope (CE) phase. Here we present spectroscopy of the Necklace taken near light-curve minimum that for the first time reveals a carbon-rich (C/O > 1) companion, a carbon dwarf, in a post-CE central star. As unevolved stars do not produce carbon, the chemical enhancement of the secondary can only be explained by accretion from the primary. Accretion most likely happened prior to the CE phase via wind accretion as not enough material can be accreted during the short CE phase. The pair of jets in the Necklace, which are observed to be older than the PN, are therefore likely to have been launched from an accretion disc around the companion during this early accretion phase. This discovery adds significant weight to the emerging scenario that jets in post-CE PNe are primarily launched by an accretion disc around a main-sequence companion before the CE phase.