The Old Covenant is the Law that was given to Israel as stated in Exodus 19:1-8 and referenced in Jeremiah 31:32.
Exodus 19:1-8 KJV
19 In the third month, when the children of Israel were gone forth out of the land of Egypt, the same day came they into the wilderness of Sinai. 2 For they were departed from Rephidim, and were come to the desert of Sinai, and had pitched in the wilderness; and there Israel camped before the mount. 3 And Moses went up unto God, and the Lord called unto him out of the mountain, saying, Thus shalt thou say to the house of Jacob, and tell the children of Israel; 4 Ye have seen what I did unto the Egyptians, and how I bare you on eagles’ wings, and brought you unto myself. 5 Now therefore, if ye will obey my voice indeed, and keep my covenant, then ye shall be a peculiar treasure unto me above all people: for all the earth is mine: 6 And ye shall be unto me a kingdom of priests, and an holy nation. These are the words which thou shalt speak unto the children of Israel. 7 And Moses came and called for the elders of the people, and laid before their faces all these words which the Lord commanded him. 8 And all the people answered together, and said, All that the Lord hath spoken we will do. And Moses returned the words of the people unto the Lord.
Jeremiah. 31:31-33 KJV
31 Behold, the days come, saith the Lord, that I will make a new covenant with the house of Israel, and with the house of Judah: 32 Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the Lord: 33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the Lord, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.