It's possible that this is true. It is said that Faramir "told his tale..." So perhaps this did include the details of the Ring, but maybe not, since Denethor shortly says
Quote:
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I can see and hear, as was my wont; and little of what you have half said or left unsaid is now hidden from me. I know the answer to many riddles.
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Presumably he is referring to the riddle about the discovery of Isildur's Bane and the Halfling, but it still isn't 100% clear whether Denethor knew this all along, or whether he guessed the riddle then. In any case, there is the implication that Faramir did not tell the whole story, or there would be nothing left unsaid...