( continues...) me ever again, you shall be cast yourself into the Fire of Doom."
--"Mount Doom", The Return of the King
Standing at the Crack of Doom, Frodo was wearing the Ring and overmastered by it. He was in his last need, attacked again by Smeagol. When Gollum bit off the Ring-finger, the Ring remained on Frodo's finger so that it continued to see Frodo as it's wearer and his commands to still be in effect. The Ring did not seem to "realize" that it was no longer connected to Frodo and therefore was unaware of its own danger when it, obeying Frodo's command, sent Gollum over the precipice into the Fire of Doom.
Gollum had also promised that he would not let Sauron have the Ring, and Sauron had just become aware of its presence and the danger when Frodo put on the Ring by the Crack of Doom. All his thought was bent immediately to it, taking him from all other pursuits and sending all the Ringwraiths to the Mount of Doom. They could not have escaped the Ring's taking by Sauron except by its unmaking.
Thus the oath sworn to Frodo on the Ring and his additional commands worked together so that Gollum was forced to become the tool for the destruction of the Ring and destroy himself the moment that Frodo was stopped by the Ring from completing the quest, claiming it.
Add in that Bilbo bequeathed the One Ring to Frodo and let him take the terrible mission to carry the Ring to Mount Doom because he understood his own infirmities of age, and we have a third Ringbearer who took part in its destruction.
SamWise, as well, may be added to that number bringing it to four, as the Ringbearer who voluntarily returned the Ring to Frodo, and who selflessly aided Frodo's quest, even to the point of giving him the last of the water and carrying Frodo on his back up the mountainside.
Add in as well the oft-forgotten Deagol, who found the lost Ring deep in the waters of the Anduin, bringing the Ring back into play for its ultimate destruction. If not for him, it would not have come to Smeagol/ Gollum to hide in the roots of the mountains until it could be given, as Gandalf said (in FotR, "Shadow of the Past") it was "meant" to be, to Bilbo. From there it was inherited by the finest of the hobbits, the one who would take the Ring to its beginnings, Frodo.
Thus, thanks to the hobbit Ringbearers, the only ones able to carry the Ring for any length of time without being taken over by a desire for power, the Ring was doomed.
References: Fellowship of the Ring: "Shadow of the Past", The Return of the King: "Mount Doom"; The Two Towers : "The Taming of Smeagol", "The Black Gate is Closed"