To answer the OP headline question, I think that in the new heavens and new earth, (Which doesn't come until "after" the millennial reign), we will no longer have any awareness or knowledge of sin, specifically, but we will have known the one rule, and that those who had it "in them" to disobey it, will no longer be, and will be done away with, this time forever, and those who "did not" "have it in them" will remain...
And, the "one rule" will still remain but it will go on forever without sin (disobeying, going against it) anymore... Because, the ones in it (now, or rather, "then") just "don't have it in them" to go against that, and have absolutely no desire whatsoever, to "ever", for all time, to ever disobey the one rule, ever again... And, with no knowledge or awareness of any sin, evil, wickedness, or badness at all, and it will be free of sin forever...
God Bless!
Many of us, Myself as a primary example, I used to not have a problem with my sin, before I came to know God and got born-again, now I hate it, but I still do it sometimes, though not as much, but I hate it and have a problem with it, when I do... If it weren't for the sinful world, the sinful flesh, and the enemy, I really feel like I would be able to be completely free of sin and sinless and happy about it too... These things will be done away with when Christ comes, and with those no longer present and a problem, It is very possible not to sin anymore under those conditions, I feel...
I know that the only reason I still do sometimes do what I now hate, even after being born-again, is because of this world, the way it is, it's influence, my own sinful flesh, and the enemy... Without these, I feel like I'd be just fine and do a heck of a whole lot better, which is what it will be like in the New Millennium, a different world without any sinful aspects to it, a new fleshly body, not prone to sin, and no more enemy... Under these conditions, don't you feel like you'd do pretty well also...?
The one rule, may be, while I don't what it will be exactly, I know it will have something to do with Love and the heart, those that get to be in the New world with Christ reigning and ruling it, will all have the Love of God in their heart, and the one rule, will have something to do with that... The two commandments Jesus gave, that he said sums up the rest, and was the very nature of "Doing unto others as you would have them do unto you." Was centered in and around "Love"...
At the end of the thousand years, the enemy may try to lead us to think God does not Love us like he says he does and is not Love, or at the very least get us to doubt God's Love, or question the very nature and purpose of love in us, or something to that effect... Perhaps, trying to cause rebellion again... It will be the last time, him and those who sin because of him, will be quickly and swiftly dealt with this time and done away with forever this time...
God Bless!