Well I believe that someone having the knowledge and strength of Celebrimbor could have made new Rings had they wanted.
And because Sauron helped teach/develop the elvish smith's Ring-lore and aided in the forging of all but the three. Therefore his influence was present in all the rings, so when he made the One, he was able to gain control over the other rings and further strengthen his influence ove the rings powers and thereby gaining influence and control over the ring-bearers.
To the best of my knowledge, since Celebrimbor forged the three in secret, Sauron was not aware of them initally. But since Ring-lore (at least it seems a large part of it) came from Sauron, the One enabled him to be aware of them, and their capability's, which enabled Sauron to recognize the three's influence (and thereby undo such influence) so long as the One was available to him.
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