(Eph. 1:3,4; 2 Thess. 2:13; Rom. 5:21; Acts 13:8; Jer. 31:33)
From the Catechism, we see that the Covenant of Grace is the historical outworking of the Covenant of Redemption, wherein God chose people and gave them to Christ for salvation from every tribe, language, people, and nation. It seems also that both the Confession and the Catechism teach that it is with the elect alone that God enters into a Covenant of Grace. This is a very important distinction between the Baptists and their Paedobaptist (Presbyterian) brethren. According to Presbyterians, both believers and their natural offspring (children) are part of the Covenant of Grace (though in different senses, see below). In Reformed Baptist thought, the Covenant of Grace is made with the elect alone.
One difference that has clearly come to me between the Baptists and the Presbyterians in the 17th century is their idea or non-idea of the administration of the Covenant of Grace. What did they mean by "administration"? The Westminster Confession 7:5 lays it out:
This covenant [the Covenant of Grace] was differently administered in the time of the law, and in the time of the gospel: under the law it was administered by promises, prophecies, sacrifices, circumcision, the paschal lamb, and other types and ordinances delivered to the people of the Jews, all fore-signifying Christ to come, which were for that time sufficient and efficacious, through the operation of the Spirit, to instruct and build up the elect in faith in the promised Messiah, by whom they had full remission of sins, and eternal salvation, and is called the Old Testament.
What they meant by "administration" is that the substance of the all the covenants in the OT are the same, namely, the Covenant of Grace, but the administration of the particular covenants is different. The substance is the same, but the (outward) form is different. This distinction justifies the practice of infant baptism when understanding their position on the Covenant of Grace...it kind of makes sense. If the Abrahamic Covenant was an administration of the Covenant of Grace and it had the sign of circumcision, which was administered to both Jacob and Esau when they were infants, then it makes sense that if the New Covenant has the same substance as the Abrahamic Covenant. This point would be carried over to the New Covenant and there we would also baptize our infants since they are part of the covenant just like in Abraham's time. Circumcision is replaced by baptism in the New Covenant administrations of the Covenant of Grace. The outward form is different, but the substance is still the same. Therefore, the promise also of "you and your seed" applies to believers in the New Covenant and their natural offspring. Such is the reasoning of our Presbyterian brethren. With such thinking, I can see some possibility of infant baptism being right, but there is more that needs to be examined before declaring infant baptism biblical. See chapter 29 for more on baptism and the Covenant of Grace.
This is not only the Presbyterian understanding but even some Reformed Baptists' understanding. I think Richard Barcellos was right in observing that many Reformed Baptists assumed a Covenant Theology like their Presbyterian brethren, while not looking at the distinct Covenant Theology of our Baptist forefathers. In a recent Reformed Baptist book this idea was promoted:
It is absolutely imperative to understand ...