When The Hobbit was first published there was no connection between Sauron and the Ring.
Sauron did exist (at the time of writing he was called Thu rather than Sauron, but the character was the same), Tolkien had always intended that he was the Necromancer, but the concept of the Rings of Power hadn't yet emerged, and therefore nor had Sauron's connection to them.
In the second or third editions of The Hobbit two references connecting Sauron to the Ring, and reflecting the new concept of the Ring entered the text of Riddles in the Dark:
But who knows how Gollum came by that present, ages ago in the old days when such rings were still at large in the world? Perhaps even the Master who ruled them could not have said.
Whether it was.an accident, or a last trick of the ring before it took a new master, it was not on his finger.
But these were not in the original text presented in John Rateliff's History of the Hobbit.
The development of the concept of the Rings of Power and their link to Sauron can be traced in History of Middle-earth 6 and 7, and there is no question but that the emergence of Bilbo's ring as the One Ring, and the precise nature of this Ring, took place during the writing of Lord of the Rings, and after many successive drafts, particularly of the chapter that became The Council of Elrond.