I agree with Eomer. Sauron believed that Aragorn had the ring. There's a quote in the Silmarillion on a letter he writes to Milton Waldman...
Quote:
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If that would happen, the new posessor could (if sufficiently strong and heroic enough by nature) challenge Sauron, become master of all that he had learned or done since the making of the One Ring, and so overthrow him and usurp his place. This was the essential weakness he had introduced into his situation in his effort (largely unsuccessfully) to enslave the Elves, and in his desire establish control over the minds and wills of his servants. There was another weakness: if the One was actually unmade, annhilated, then its power would be dissolved, Sauron's own reduced to a shadow, a mere memory of malicious will. But that he never contemplated nor feared. The Ring was unbreakable by any smithcraft less than his own. It was indissoluble in any fire, save the undying subterraneans fire where it was made-and that was unapproachable, in Mordor. Also, so great was the Ring's power of lust, that anyone who used it became mastered by it; it was beyond the strength of any will (even his own) to injure it, cast it away, or neglect it.
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Sauron never feared the destruction of the Ring (hence why he didn't have really any concern for someone sneaking into Mordor going to Mount Doom and dropping it in). He believed no one could destroy it, nor would anyone want to destroy it.
The thing he feared was someone else using the Ring to try to overthrow him. He believed Aragorn had the ring, he believed naturally Isildur's heir would come to try to use it against him and overthrow him. The Mouth of Sauron went out to try to get Aragorn and co. to bust what they were doing, by telling them they found their spies and they suffered a horrible death. That's all Sauron thought the Hobbits were, spies out of Minas Tirith, he didn't think they had the ring.