What Apostles-Creed affirming groups claim priesthood of Melchizidek?

Question

I was looking at the online biography (perhaps an AOL or MySpace biography) of the friend of a relative some 20 years ago. I was alarmed that my relative was associated with this person, based on some very immoral vices the person was proudly describing on their page. One point that shocked me was that this person said they were "a priest of the order of Melchizidek." It was the only religious statement on the page.

Q: Do any orthodox Christian groups (perhaps defined here as affirming the Apostle's Creed) use such language to describe themselves? Or would this be some sort of cult?

Answer 1

The LDS Church uses this phrase. They divide the priesthood into Melchizedek Priesthood and Aaronic Priesthood.

Melchizedek Priesthood is described on their site as follows:

“There are, in the church, two priesthoods, namely, the Melchizedek and Aaronic” (D&C 107:1). The Melchizedek Priesthood, which is “after the Order of the Son of God” (D&C 107:3), is the greater of these. It “holds the right of presidency, and has power and authority over all the offices in the church” (D&C 107:8). It also holds “the keys of all the spiritual blessings of the church” (D&C 107:18). It is named after a great high priest who lived during the time of the prophet Abraham (see D&C 107:2-4; see also Alma 13:14-19).

The offices of the Melchizedek Priesthood are Apostle, Seventy, patriarch, high priest, and elder. The President of the High Priesthood is the President of the Church (see D&C 107:64-66).

Men in the Church must be worthy Melchizedek Priesthood holders in order to receive the temple endowment and be sealed to their families for eternity. They have the authority to administer to the sick and give special blessings to family members and others. With the authorization of presiding priesthood leaders, they can bestow the gift of the Holy Ghost and ordain other worthy men to offices in the Aaronic and Melchizedek Priesthoods.

When a man receives the Melchizedek Priesthood, he enters into the oath and covenant of the priesthood. He covenants to be faithful, magnify his calling, “give diligent heed to the words of eternal life,” and “live by every word that proceedeth forth from the mouth of God.” Those who keep this covenant will be sanctified by the Spirit and receive “all that [the] Father hath.” (See D&C 84:33-44.)

Specifically, John Taylor is quoted as saying (Emphasis mine)

Seventh—That while the power of the higher, or Melchizedek, is to hold the keys of all the spiritual blessings of the church; to have the privilege of receiving the mysteries of the kingdom of heaven, to have the heavens opened to them, to commune with the general assembly and church of the firstborn and to enjoy the communion and presence of God the Father, and Jesus the Mediator of the new covenant, and to preside over all the spiritual officers of the church, yet the presidency of the high priesthood, after the order of Melchizedek, have a right to officiate in all the offices in the church, both spiritual and temporal.

Answer 2

According to Scripture (the letter to the Hebrews), it is Jesus Himself who is a priest like Melchizedek.

The original Melchizedek was both king of the small city-state of Salem (which later because Jerusalem) and the high priest of its religion. (They worshipped El Elyon, God Most High, which title Abraham recognized as another name for YHWH.) There is only ever one "priest in the order of Melchizedek" at a time. There can't be more than one. Some scholars suggest that Melchizedek was actually a pre-incarnation appearance of Jesus.

Here's what Hebrews chapter 7 says:

7 This Melchizedek was king of Salem and priest of God Most High. He met Abraham returning from the defeat of the kings and blessed him, 2 and Abraham gave him a tenth of everything. First, the name Melchizedek means “king of righteousness”; then also, “king of Salem” means “king of peace.” 3 Without father or mother, without genealogy, without beginning of days or end of life, resembling the Son of God, he remains a priest forever.

4 Just think how great he was: Even the patriarch Abraham gave him a tenth of the plunder! 5 Now the law requires the descendants of Levi who become priests to collect a tenth from the people—that is, from their fellow Israelites—even though they also are descended from Abraham. 6 This man, however, did not trace his descent from Levi, yet he collected a tenth from Abraham and blessed him who had the promises. 7 And without doubt the lesser is blessed by the greater. 8 In the one case, the tenth is collected by people who die; but in the other case, by him who is declared to be living. 9 One might even say that Levi, who collects the tenth, paid the tenth through Abraham, 10 because when Melchizedek met Abraham, Levi was still in the body of his ancestor. Jesus Like Melchizedek

11 If perfection could have been attained through the Levitical priesthood—and indeed the law given to the people established that priesthood—why was there still need for another priest to come, one in the order of Melchizedek, not in the order of Aaron? 12 For when the priesthood is changed, the law must be changed also. 13 He of whom these things are said belonged to a different tribe, and no one from that tribe has ever served at the altar. 14 For it is clear that our Lord descended from Judah, and in regard to that tribe Moses said nothing about priests. 15 And what we have said is even more clear if another priest like Melchizedek appears, 16 one who has become a priest not on the basis of a regulation as to his ancestry but on the basis of the power of an indestructible life. 17 For it is declared:

“You are a priest forever, in the order of Melchizedek.”[a]

18 The former regulation is set aside because it was weak and useless 19 (for the law made nothing perfect), and a better hope is introduced, by which we draw near to God.

20 And it was not without an oath! Others became priests without any oath, 21 but he became a priest with an oath when God said to him:

“The Lord has sworn and will not change his mind: ‘You are a priest forever.’”[b]

22 Because of this oath, Jesus has become the guarantor of a better covenant.

23 Now there have been many of those priests, since death prevented them from continuing in office; 24 but because Jesus lives forever, he has a permanent priesthood. 25 Therefore he is able to save completely[c] those who come to God through him, because he always lives to intercede for them.

26 Such a high priest truly meets our need—one who is holy, blameless, pure, set apart from sinners, exalted above the heavens. 27 Unlike the other high priests, he does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself. 28 For the law appoints as high priests men in all their weakness; but the oath, which came after the law, appointed the Son, who has been made perfect forever.

Sects which claim a Melchizedek priesthood for some of their clergy are talking unbiblical nonsense. Their clergy are mortal men who age and die. They are not "without beginning of days nor end of life".

So Jesus is the one and only Melchizedek-type priest, and the temporary priesthood of Aaron (which was central to the Old Covenant between God and the 12 tribes of Israel) is gone.