Did the Council of Elrond know that destroying the Ring would destroy Sauron?
"The Dark Lord Sauron forged in secret a master ring, to control all others. And into this ring he poured his cruelty, his malice, and his will to dominate all life. One Ring to rule them all."
—Galadriel regarding Sauron and the forging of the One Ring.
They knew he poured so much of himself into the ring, that destroying it would destroy him with it.
After a brief period of renewed war, Elendil and his people allied with the Elven-king, Gil-Galad to create the Last Alliance, and together fought Sauron. They finally defeated his armies at the Battle of Dagorlad, and sieged Barad-dûr for seven years. Finally, Sauron himself came forth and dueled both Elendil and Gil-galad. He slew them both but Isildur, son of Elendil, took up his father's broken sword, Narsil and cut the One Ring from Sauron's finger, after which Sauron's spirit fled again in ruined form.
The Council of Elrond knew that much of Saurons power came from the One Ring -
Sauron without the Ring lost his ability to take any physical form for a long time, and possibly retreated into the Far East until the time was right.
While Sauron did have a physical form, he remained much weakened without the One Ring and remained hidden in the shadows, directing his armies from afar.
Frodo was able to reach Mount Doom, and upon putting on the Ring, Sauron suddenly became aware of him. Though enraged, he was suddenly gripped with terror, realizing his own folly, and frantically sent the Ringwraiths towards the mountain to retrieve the Ring. He was too late however, and Gollum, after taking the Ring from Frodo, slipped into the Cracks of Doom.
With his source of power destroyed, Sauron was utterly defeated. Barad-dûr fell and his armies were destroyed or scattered, bereft of the driving will behind their conquest, and Mordor itself was shaken to its core as Orodruin belched fire. And, all Eight remaining Nazgul had been in combat with the Great Eagles, and lost, now being helplessly drawn towards the gale of lava on their Fellbeasts, and were consumed.