When Paul speaks of "the ministry of death," he refers to the administration of the Old Covenant rather than the Ten Commandments. The Levitical priesthood, a carnal priesthood based on physical descent from Levi, administered the Old Covenant. This covenant provided no promise of eternal life and no means for sinners to receive forgiveness because "it is not possible that the blood of bulls and goats could take away sins" (Hebrews 10:4). Therefore, the people lived and died under the condemnation of the law, and "the wages of sin is death" (Romans 6:23).
Another reason why Paul refers to the Old Covenant as "the ministry of death" is that God required the Levitical priesthood to execute those who transgressed certain laws. God's law mandates the death penalty for certain sins like murder and dishonoring parents (Exodus 21:12-17), Sabbath-breaking (Exodus 31:14-15) and certain sexual sins (Leviticus 20:10-13). The priests were responsible to enforce the death penalty by actually putting such transgressors to death in the proscribed manner. In this sense, the Old Covenant ministry was indeed a "ministry of death."
However, why did Paul say that the "ministry of death," the administration of the Old Covenant, was "written and engraved on stones"? Was it not the Ten Commandments that God wrote on two stone tablets? Even though the Ten Commandments were not the covenant itself (a covenant is simply an agreement between two parties), they were the terms of the covenant. Because the Ten Commandments constituted the part of the agreement between God and Israel that the Israelites agreed to keep, the Old Covenant became synonymous with the Ten Commandments. In Deuteronomy 4:13 Moses writes, "So He declared to you His covenant which He commanded you to perform, that is, the Ten Commandments; and He wrote them on two tables of stone." To put it another way, "keeping the Old Covenant" was the same as "keeping the Ten Commandments."
A paraphrase of the first eleven words of II Corinthians 3:7 helps to clarify what Paul means: "But if the administration of the Old Covenant, [the terms of which were] written and engraved on stones. . . ." The Ten Commandments undergirded all the laws that God gave to Israel—laws that the Israelites could not keep. The responsibility to teach these laws to Israel and enforce penalties for disobedience, including the death penalty, fell to the priests.
Therefore, if perfection were through the Levitical priesthood (for under it the people received the law), what further need was there that another priest should rise according to the order of Melchizedek, and not be called according to the order of Aaron? (Hebrews 7:11)
When Moses went up Mount Sinai the second time to receive the Ten Commandments, he wrote God's statutes and judgments in a book, and God wrote the Ten Commandments on two tables of stone. This, in essence, finalized the "contract" that God made with Israel.
Then the LORD said to Moses, "Write these words, for according to the tenor of these words I have made a covenant with you and with Israel." So he was there with the LORD forty days and forty nights; he neither ate bread nor drank water. And He wrote on the tablets the words of the covenant, the Ten Commandments. (Exodus 34:27-28)
Verses 29-35 then describe how Moses face shone when he delivered the Ten Commandments and the book of the law to Israel.
So what is passing away? Hebrews 8:13 provides the answer: "In that He says, 'A new covenant,' He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away." The Old Covenant and the Old Covenant ministry, the Levitical priesthood, are passing away, not the Ten Commandments!