"Who also hath made us able ministers of the new testament (diatheke); not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life" (2 Cor.3:6; KJV).
When we look at the words describing the New
Diatheke (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).
John Calvin certainly understood that Paul was speaking of the ministry to preach the gospel, writing the following:
"'Not of the letter but of the spirit'...There is, however, no doubt, that by the term 'letter,' he means the Old Testament, as by the term 'spirit' he means the gospel; for, after having called himself a 'minister of the New Testament,' he immediately adds, by way of exposition, that he is a 'minister of the spirit,' and contrasts the letter with the spirit" [emphasis added] (John Calvin, Commentary on Corinthians - Volume 2, Christian Classics Ethereal Library).
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).
Therefore, we can understand that when Paul uses the word
diatheke at 2 Corinthians 3:6 the meaning of that word is the "gospel."
It's easy to understand how a person can minister the gospel but how does a person minister a covenant, specifically the New Covenant?