Who Owns the Cat? (Who Stole What from Whom? Part IV)

The cat belonged to one of three girls—Annabelle, Betsy, or Cynthia. Annabelle claimed that Betsy doesn't own the cat, and Betsy claimed that Cynthia owns the cat. Now, it so happens that the the owner of the cat always tells the truth and is the only one of the three girls who ever tells the truth.
Who owns the cat?

(Source: King Arthur in Search of His Dog and Other Curious Logic Puzzles by Raymond Smullyan)

Suppose that Annabelle is not the owner. This would imply that her claim that Betsy doesn't own the cat is a falsehood. This would in turn imply that Betsy is the owner of the cat and therefore speaks truth. But if Betsy were the owner of the cat and accordingly a truth-teller, she would not claim that Cynthia owns the cat, which is a contradiction. Therefore Annabelle must be the owner of the cat.