Big spherulite structure and high crystallinity are the two main drawbacks of poly(butylene succinate) (PBS) and hinder its application. In this work, a new type of copolyester poly(butylene succinate-co-butylene acetylenedicarboxylate) (PBSAD) is synthesized. With the incorporation of acetylenedicarboxylate (AD) units into PBS chains, the crystallization temperature and crystallinity are depressed by excluding AD units to the amorphous region. In contrast, the primary nucleation capability is significantly strengthened, without changing the crystal modification or crystallization kinetics, leading to the recovery of total crystallization rate of PBSAD under the same supercooling condition. The existence of specific interaction among AD units is found to be crucial. Although it is too weak to contribute to the melt memory effect at elevated temperature, the interaction continuously strengthens as the temperature falls down, and the heterogeneous aggregation of AD units keeps growing. When the aggregating process reaches a certain extent, it will induce the formation of a significant amount of crystal nuclei. The unveiled nucleation mechanism helps to design PBS copolymer with good performance.