Was the Council of Elrond aware that destroying the ring would vanquish Sauron?

Question

In Fellowship of the Ring, were those at the Council of Elrond aware that destroying the One Ring would defeat Sauron? I'm guessing that they must have known this because nobody seemed surprised that Sauron just fell apart once the Ring was cast into Mount Doom, so why did no one mention this at the meeting?

Boromir advocated using the Ring to defeat Sauron head on so it would seem perfectly appropriate to bring up that destroying the Ring would achieve the same outcome.

I don't remember much about the scene from the book so this is about the movie. However, I would also love to hear answers based on the book and other written sources.

Answer 1

Yes

Gandalf, a member of the White Council, discusses this in the books:

"If it [the One Ring] is destroyed, then he [Sauron] will fall, and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed."

Return of the King, ch. 9 The Last Debate

H/T Adam V on Sci-Fi Stack Exchange

Answer 2

I don't believe that there was any mention in the books that destroying the Ring would destroy Sauron and no mention at the Council of Elrond. The goal was to prevent Sauron from obtaining the One Ring, which he would use to conquer middle-earth. In The Fellowship of the Ring, chapter 2 - The Shadow of the Past, Gandalf tells Frodo that the only way to destroy the ring is "to find the Cracks of Doom in the Depths of the Orodruin, the Fire-mountain, and cast the Ring in there"

Answer 3

The one ring was the only thing that kept him alive after he lost his physical body in the first great battle at mt. doom. Since the Istari knew Sauron was only banished, not dead, but Elrond didn't kill Isildur for walking away with the ring, they knew that his spirit could be vanquished destroying the one ring.

Answer 4

It's said that Sauron put al his power and evilness into the ring so it was known that te ring and sauron's coming back were deeply related and that his power depended on his proximity to the ring. So, in my opinion, cause it really isn't written is that they knew it because of the clues that they left when they told the story.