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Yes. The law specifically says:
§ 51-1. Requisites of marriage; solemnization.
A valid and sufficient marriage is created by the consent of a male and female person who may lawfully marry, presently to take each other as husband and wife, freely, seriously and plainly expressed by each in the presence of the other, either:
(1) a. In the presence of an ordained minister of any religious denomination, a minister authorized by a church, or a magistrate; and
b. With the consequent declaration by the minister or magistrate that the persons are husband and wife; or
(2) In accordance with any mode of solemnization recognized by any religious denomination, or federally or State recognized Indian Nation or Tribe.
Marriages solemnized before March 9, 1909, by ministers of the gospel licensed, but not ordained, are validated from their consummation.
You don't need to register to perform a wedding ceremony. You do need to make sure that the couple has a license, as you cannot perform a ceremony without a marriage license from the couple.
So long as you're an ordained minister of "any denomination", then that would legally suffice.
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