No, perhaps you're confusing JB paste with copper cleaner. The copper cleaner cleans down to the steel of the barrel and nothing more. JB paste is an actual rubbing compound containing grit (albeit fine) that is grinding the steel of the barrel away. So it will come out black forever.
With a new barrel, I like to follow my copper cleaning with a few passes of JB until the rifle is broken in. (that's a subjective term but to me it means cleaning the copper out isn't a great chore) I run the JB thru the bore until it feels "slick" and then swab it out using Kroil or Hoppes followed by brake cleaner (the other day, I heard Windex is good) and then a patch of Kroil followed by two dry patches.