TT #42 · Parshas Tazria · Aliyah 1 · Passuk 6
English: On the completion of her period of purification, for either son or daughter, she shall bring to the priest, at the entrance of the Tent of Meeting, a lamb in its first year for a burnt offering, and a pigeon or a turtledove for a purgation offering.
Chabad.org — Chumash, English translation + Rashi for Vayikra 12:6
Torah Temimah cites Nazir 64b for the rule distinguishing a birth on the day the period is completed from a birth during that period.
E-Daf — Nazir 64b · Shas.org · Sefaria — Nazir 64b, vowelized text and translation
English: “When the days of her purification are completed” — on the day of completion she brings an offering; within the period of completion she does not.
Textual limitation
When a woman gives birth and then has another birth or miscarriage during the first birth’s purification period, the additional event does not require a separate offering. If it occurs on the day the period is completed, the rule can require a separate offering. The baraita distinguishes these cases; it is not a general rule that an ordinary childbirth offering is brought only on that single day.