“Ye” was pronounced “the.”
“Ye” is a 16th century substitution of a “y” for an Old English character known as the thorn (“þ”), originally a Germanic rune that represented the interdental th sound. In late 15th century, early English printers, whose types were still founded on the continent, did not have the þ so they substituted the y, which read enough like a þ when set in type. It dropped from usage as “th” gained favor, but was revived in the 19th century as a self-conscious antiquarianism.
Note: The “ye” in the carol, God Rest Ye Merry Gentlemen was not an article, it was a familiar second person pronoun (“thou” would have been the singular), and thus, another word altogether.