It would seem that based on the כלל of כל העומד that you can't use it. Being that the Ring is to be destroyed, we may apply כל העומד לשרוף כשרוף דמי, and say that the ring is כתותי מכתת שיעוריה. If so, the Ring is in essence worthless.
Furthermore, we פסקין in מסכת סנהדרין that someone who worships one's self is עובר on עבודה זרה, and being that the Ring was basically a form of self-worship on the part of Sauron, the ring is אסור בהנאה as עבודה זרה.
In response to Daniel above, it is theoretically possible that the Ring will be stolen and acquired by the thief, but being that A) Sauron was not מייאש from the Ring, and B) it is not possible to affect any שינוי on the Ring, as is proven from the time that Gandalf threw it into the fire in the chapter "A Shadow from the Past", is is not possible that anyone was קונה through שינוי. When Frodo declared that it was his on the Fiery Mountain, it is possible that he was קונה through שינוי רשות, bt at that point it is obviously כל הומד לשרוף כשרוף דמי, if he was קונה or not is a moot point.