The word order in that sentence is less important, because it uses old pronoun "thee" as opposed to "thou" (analogous to German "dich" vs "du"), thus indicating

The word order in that sentence is less important, because it uses old pronoun "thee" as opposed to "thou" (analogous to German "dich" vs "du"), thus indicating the subject and object. "With this ring, thee I wed" "I wed thee, with this ring" ... "Thee, with this ring, I wed" (???) When you use "you", suddenly there is only one obvious order.