Gollum takes the Ring from Frodo by biting off the finger bearing the Ring.
Frodo already knew that Gollum was Smeagol because Gandalf told him earlier in the Fellowship of the Ring. When Gollum hears his old name, he realizes who he really is and that his master Frodo is the one who takes care of him now and no longer needs "Gollum".
Gandalf told Frodo about Gollum's history while they were at Bag End then later in the story Frodo told Gollum his true name.
Frodo realized how the Ring of Sauron was destroying him and his nature with less than a year of serious exposure and could empathize with a creature like Gollum who was once a Hobbit but had 500 years over which the Ring of Sauron destroyed him. Frodo's understanding of Gollum's pain derived from the Ring of Sauron is what leads to this pity.
In reality, Frodo was defeated by Sauron, the One Ring's power overcame Frodo in the end and he announced his ownership of the ring, this drew Sauron's attention to Mount Doom and he ordered a Nazgul there to take the One Ring from Frodo and return it to its Maker - Sauron. Frodo would not have been able to fight a Nazgul and win, even with the One Ring to assist him as the ring wanted to return to its Maker and would not have assisted Frodo.All was now lost, except for one minor detail. Perhaps a half hour before arriving at the Cracks of Doom, Gollum had attacked Frodo and after Sam and Frodo had fought him off, Frodo in anger (perhaps without even realizing it) cursed Gollum, by the power of the ring, that "if Gollum ever touched Frodo again he would fall in the Cracks of Doom".Only moments before the Nazgul could arrive to take the ring, Gollum jumped Frodo, wrestled with him, bit off his ring finger (if that's not touching, I don't know what is). This activated Frodo's curse on Gollum by the power of the ring, causing Gollum to fall in the Cracks of Doom while celebrating taking "his" ring back, the ring fell with him and was thereby destroyed, thus fulfilling the quest of the Fellowship of the Ring.With the One Ring gone all Sauron's power was also gone, the Nazgul turned to smoke, Baradur collapsed, his armies scattered without guidance, and Sauron himself probably followed his master Morgoth into the void.
When he falls in the "Cracks of Doom" with Frodo's ring finger.
Gollum simply isn't trustworthy. Faramir realized the importance of destroying the Ring, especially for the safety of his beloved people. He naturally distrusted anything that might stand in the way of Frodo's quest. He did not know Gollum's history, so did not have pity on him as Frodo did. He undoubtedly saw him as a sneaky, greedy thief- which he was, of course. Unlike Faramir, I have pity on Gollum/Smeagol and consider him my 3rd favorite character. :)
From Sauron's hand to Isildur to the bottom of the Anduin River to Dėagol to Smėagol (Gollum) to Bilbo and thence to Frodo.
He leads frodo an Sam towards mordor and slowly turns frodo against Sam. Then when he falls off a cliff after a fight with frodo, we don't see his for a while until Sam and frodo are climbing up the slopes of mount doom. He then attacks them and tries to take the ring. Eventually Sam and frodo escape and are standing on the edge of the lava about to drop the ring into the volcano when gollum hits Sam on the head with a rock and then bites frodos finger off and claims the ring for himsefl, jumping around with joy. THEN frodo attacks gollum an they fight for a bit before they both fall off the edge. Frodo manages to hold onto the rock but gollum, who is holding the ring, falls into the lava and dies.
His finger. Gollum bit it off to get the ring.
Gollum and Smaug are introduced in The Hobbit; Frodo is introduced in The Lord of the Rings.
The One Ring changed hands several times: It was forged by Sauron Isildur cut the Ring from Sauron's hand. He was shot by orcs when the Ring slipped into the Anduin River Deagol took the Ring from the Anduin Smeagol, later Gollum strangled Deagol to get the Ring, and kept it for over 500 years The Ring was picked up by Bilbo Baggins in a dark cave At the urging of Gandalf, Bilbo passed the Ring to Frodo Baggins Sam took the Ring from Frodo when Frodo was stung by Shelob Sam freely returned the Ring to Frodo Gollum bit the Ring and the finger it was on off Frodo and fell into the fires of Mt. Doom with it. The One Ring was destroyed.
Gollum took it from him and Frodo pushed him into Mt Doom