A "covenant" is not the same thing as a "testament." It was the Lord Jesus' death that provided the blessings of His last will and testament.
Today those in the Body of Christ are saved by the new testament, the gospel, and not by the new covenant promised to the house of Israel and the house of Judah. Let us look at this verse which speaks of the New Testament:
"Who also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life" (2 Cor.3:6).
When we look at the words describing the New Testament (the spirit giveth life) we can understand that the ministry in question is preaching the gospel. That is the ministry which Paul speaks of here:
"But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God" (Acts 20:24).
In the
Jamieson-Fausset-Brown Bible Commentary we read the following remarks on 2 Corinthians 3:6:
"spirit giveth life--The spirit of the Gospel when brought home to the heart by the Holy Spirit, gives new spiritual life to a man (Ro 6:4, 11)" [emphasis added] (Jamieson-Fausset-Brown Bible Commentary, Commentary at 2 Corinthians 3:6).