The author of the Letter to the Hebrews clearly identifies Christ to be this
priest:
For it is evident that our Lord was descended from Judah, and in
connection with that tribe Moses said nothing about priests. This becomes
even more evident when another priest arises in the likeness of
Melchizedek, who has become a priest, not according to a legal
requirement concerning bodily descent but by the power of an
indestructible life. For it is witnessed of him, Thou art a priest forever,
after the order of Melchizedek (7:14-17).
After the order of Melchizedek means in the manner of Melchizedek.
Melchizedek brought forth bread and wine and sacrificed them by offering
them to Abraham to eat. Christ is a priest after this manner by offering His
Body and Blood under the veil of bread and wine for us to eat.
The Book of Daniel chapter 9 speaks of the end of the Jewish priesthood
and its sacrifices:
After the sixty-two weeks, an anointed one shall be cut off and shall have
nothing, and the troops of the prince who is to come shall destroy the city
and the sanctuary. Its end shall come with a flood, and to the end there
shall be war. Desolations are decreed. He shall make a strong covenant
with many for one week, and for half of the week he shall make sacrifice
and offering cease; and in their place shall be an abomination that
desolates, until the decreed end is poured out upon the desolator (vv. 26-
27).
The Jewish priesthood and sacrifices would be replaced by Gentile ones as
predicted by the Prophet Malachi:
I have no pleasure in you, says the Lord of hosts: and I will not receive a
gift of your hand. For from the rising of the sun, even to the going down,
my name is great among the Gentiles, and in every place there is sacrifice,
and there is offered to my name a clean oblation: for my name is great
among the Gentiles, saith the Lord of hosts (Mal. 1:10-11).
Malachis words found fulfillment in the worship of the early Christians: