"The day of the Nativity of the Mother of God is a day of universal joy, because through the Mother of God, the entire human race was renewed, and the sorrow of the first mother, Eve, was transformed into joy."
– St. John Damascene
The birth of the Blessed Virgin Mary has been celebrated as a universal liturgical feast at least from the sixth century. Its origin can be traced to the occasion of the consecration of a church in Jerusalem just inside St. Stephen’s Gate, near the Pool of Bethesda — a church built on the traditional site of the house of Ss. Joachim and Anne. There they lived as husband and wife. Their love for God and for each other brought them the precious gift of their daughter, Mary. From the earliest years, the Church venerated the place where Mary was born, and the liturgical remembrance began to spread beyond Jerusalem. Within a few years it was celebrated in Rome, having been introduced by monks from the East, and the celebration included a procession to the Basilica of St. Mary Major.
Although the actual date of Mary’s birth isn’t known, the Church settled on September 8th, and the celebration Solemnity of the Immaculate Conception was fixed on December 8th, as the date corresponding to nine months before the celebration of her Nativity.
The two feasts commemorating Mary's conception and her birth can be seen as forming a kind of bridge between the Old Testament and the New Testament. With the conception and birth of the Blessed Virgin, God completed the new Ark – the living Temple – in which He would dwell. Because of that there is no more need for the old Temple, and through Mary, Jesus the Incarnate God has come to us to incorporate us into the New Israel.
We beseech thee, O Lord, pour into our hearts the abundance of thy heavenly grace; that, like as the child bearing of the Blessed Virgin Mary was unto us thy servants the beginning of salvation, so the devout observance of her Nativity may avail for the increase of our peace; through Jesus Christ our Lord, who liveth and reigneth with thee and the Holy Spirit, one God, for ever and ever. Amen.
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Actually the celebration of Mary's Conception is the older feast. The Eastern Church celebrated the feast of the Conception of the Most Holy and All Pure Mother of God perhaps as early as the 5th century in Syria. The date of the feast was December 9th, still the date in the Orthodox Church. The feast of Mary's Nativity came later.
The history of the various feasts is fascinating, and a specific time-line is, at best, sketchy. My words about fixing the dates was not intended to indicate which came first; only that they are fixed nine months apart.