The synthesis of high‐molecular‐weight polyesters via the ring‐opening alternating copolymerization (ROAC) of cyclic anhydrides and epoxides using organocatalysts remains a major challenge. In this context, the organic cyclic trimeric phosphazene base (CTPB) was used as an efficient catalyst for the ROAC of phthalic anhydride (PA) with epoxides. It was found that the activity and PA conversion dramatically increased with the addition of triethyl borane (TEB) as a cocatalyst, even a small amount of TEB, for example, 5 mol % relative to CTPB. The molecular structures of obtained polyesters were characterized by nuclear magnetic resonance (1H NMR, 13C NMR) and matrix‐assisted laser desorption/ionization time‐of‐flight (MALDI TOF), which demonstrated the formation of perfectly alternating polyesters with unimodal and narrow molecular weight distributions. Different molecular weight polyesters (up to 118.5 kg mol−1) were facilely prepared by reducing the loading of CTPB and TEB, while the polymerizations were fast (turnover frequency, TOF up to 500 h−1). Thermal analysis of the resulting polymers indicated that the alternating polyesters were amorphous, and their Tgs increased with the increment of molecular weights. © 2020 Wiley Periodicals, Inc. J. Polym. Sci. 2020, 58, 803–810