In a register office it will be the registrar.
And if you live in Pennsylvania or Maryland, which have strong Quaker populations, in which there is no "minister" you can join yourselves in marriage, as long as you have two witnesses.
I did that. Our friends witnessed us "unite ourselves in marriage." Pretty cool, huh?
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.