Because of St Ambrose's ambiguity in this infamous quote, it forces us to look elsewhere in his writings for clarification.
St. Ambrose, De mysteriis, 390-391 A.D.:
“You have read, therefore, that the three witnesses in Baptism are one: water, blood, and the spirit; and if you withdraw any one of these, the Sacrament of Baptism is not valid. For what is water without the cross of Christ? A common element without any sacramental effect. Nor on the other hand is there any mystery of regeneration without water: for ‘unless a man be born again of water and the Spirit, he cannot enter the kingdom of God.’ [John 3:5] Even a catechumen believes in the cross of the Lord Jesus, by which also he is signed; but, unless he be baptized in the name of the Father and of the Son and of the Holy Spirit, he cannot receive the remission of sins nor be recipient of the gift of spiritual grace.”
St. Ambrose, The Duties of Clergy, 391 A.D.:
“The Church was redeemed at the price of Christ’s blood. Jew or Greek, it makes no difference; but if he has believed he must circuмcise himself from his sins so that he can be saved;...for no one ascends into the kingdom of heaven except through the Sacrament of Baptism.”
St. Ambrose, The Duties of Clergy, 391 A.D.:
“Unless a man be born again of water and the Holy Spirit, he cannot enter the kingdom of God.’ No one is excepted: not the infant, not the one prevented by some necessity.”
Unless one wants to make out one of the great doctors of the Church as some sort of schizophrenic, he cannot be pointed to as supporter of BOD.