How do we know Frodo's Ring is the One Ring? It turns you invisible and prolongs your life, which we know means it can't be an ordinary magic ring... because Gandalf told Frodo it can't be. We know it can't be melted in any fire less than that of Mount Doom... because Gandalf threw it into a fire that wouldn't even melt ordinary gold, and then told Frodo it can't be. We know it has writing on it which translates into 'One Ring to rule them all'... because Gandalf told Frodo that's what it says. We know it was found in the area Isildur died... because Gandalf told Frodo that's what he figured out from Gollum's tale. We know that Sauron is searching for it... because Gandalf says that's what he's doing.
The only evidence we have for anything about Frodo's Ring is that a) it turns you invisible (like any other magic ring), b) the Ringwraiths can sense it (and how do we know they can't sense any magic artefact? Oh, right, Gandalf said so), and c) it makes you want to keep it (like the Arkenstone, and the Silmarils, and basically any magic artefact). Everything else we know about it comes from Gandalf.
Oh, we also know that throwing it into Mount Doom brought down the Barad-dur. So maybe it was the One Ring? Only... the only reason we know destroying the One would destroy Sauron is that Guess Who the Grey told us so.
There is an alternative. The Ring Frodo carried had a strong connection with fire. It couldn't be heated by Frodo's hearthfire; it tried to boil the Mirror of Galadriel; Frodo described it in Mordor as 'the wheel of fire'; and, ultimately, throwing it into the largest fire in Middle-earth tore a Dark Lord's realm apart.
There is a Ring of Power with a strong fire connection - but it's not the One. It's one of the Three: Narya. And guess who had custody of Narya? That's right: Gandalf.
But we know the Three each had precious stones to mark them - oh, wait, no we don't, that's just something Gandalf told Frodo. We don't even know that Frodo's Ring is the same Ring as Bilbo had - our only source for that is Gandalf, too. (Oh, both Bilbo and Gollum thought it was - but are they jewellers, or experts in magical artefacts? I think not) Gandalf could very easily have swapped his own ring - the unadorned Ring of Fire - for Bilbo's, before Frodo ever laid eyes on it.
So what of the Ring Gandalf stole from Frodo? Simple: he was still wearing it when the White Ship set sail to the Undying Lands. That Ring could very well have been the Master Ring. And now, on the hand of a demigod equal in power to Sauron, it's on its way to Valinor...