The core-degenerate (CD) scenario has been proposed as an alternative way for the production of type Ia supernovae (SNe Ia). In this scenario, SNe Ia are formed at the final stage of common-envelope evolution from a merger of a carbon-oxygen white dwarf (CO WD) with the CO core of an asymptotic giant branch companion. However, the birthrates of SNe Ia from this scenario are still not well determined. In this work, we performed a detailed investigation on the CD scenario based on a binary population synthesis (BPS) approach. The SN Ia delay times for this scenario are mainly in the range of ∼90 Myr−2200 Myr, contributing to the observed SNe Ia with short and intermediate delay times. The Galactic birthrates of SNe Ia from this scenario range from ∼4.6×10⁻⁵yr⁻¹ to ∼4.3×10⁻⁴yr⁻¹, accounting for ∼2−14% of total SNe Ia. Especially, SNe Ia with circumstellar material from this scenario contribute to ∼1−10% of total SNe Ia, which means that the CD scenario can reproduce the observed birthrates of SNe Ia like PTF 11kx.