Counterfeit Coin With One Weighing

You have 9 visually identical coins. Exactly one of them is counterfeit -- it is either heavier or lighter than the genuine coins, but you don't know which. You have a balance scale and you may use it exactly **once**. Is it possible to *always* identify which coin is counterfeit **and** determine whether it is heavier or lighter? Give a rigorous information-theoretic argument. Specifically, count the number of distinguishable hypotheses and compare it to the number of possible outcomes of a single weighing.