Skeptics, I'm tossing you a very old chestnut indeed. This is the story of "Eureka!", and how Archimedes sent a thieving goldsmith to the executioner. Every introductory chem class re-enacts this moment in one or two of their trick questions, albeit usually without credit.
YouPhysics and Wikipedia tell the story in a way that involves using a balance underwater. Wikipedia goes to some length to discuss the plausibility of the idea. Fortunately, Wikipedia also cites the original source, which describes a different, simpler (and just as accurate) method of measuring the volume of items of equal weight underwater. To quote Marcus Vitruvius Pollio, de Architectura, Book IX (Joseph Gwilt translation):
Though Archimedes discovered many curious matters which evince great intelligence, that which I am about to mention is the most extraordinary. Hiero, when he obtained the regal power in Syracuse, having, on the fortunate turn of his affairs, decreed a votive crown of gold to be placed in a certain temple to the immortal gods, commanded it to be made of great value, and assigned an appropriate weight of gold to the manufacturer. He, in due time, presented the work to the king, beautifully wrought, and the weight appeared to correspond with that of the gold which had been assigned for it.
But a report having been circulated, that some of the gold had been abstracted, and that the deficiency thus caused had been supplied with silver, Hiero was indignant at the fraud, and, unacquainted with the method by which the theft might be detected, requested Archimedes would undertake to give it his attention. Charged with this commission, he by chance went to a bath, and being in the vessel, perceived that, as his body became immersed, the water ran out of the vessel. Whence, catching at the method to be adopted for the solution of the proposition, he immediately followed it up, leapt out of the vessel in joy, and, returning home naked, cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, εὑρηκα, (I have found it out).
After this, he is said to have taken two masses, each of a weight equal to that of the crown, one of them of gold and the other of silver. Having prepared them, he filled a large vase with water up to the brim, wherein he placed the mass of silver, which caused as much water to run out as was equal to the bulk thereof. The mass being then taken out, he poured in by measure as much water as was required to fill the vase once more to the brim. By these means he found out what quantity of water was equal to a certain weight of silver.
He then placed the mass of gold in the vessel, and, on taking it out, found that the water which ran over was lessened, because, as the magnitude of the gold mass was smaller than that containing the same weight of silver. After again filling the vase by measure, he put the crown itself in, and discovered that more water ran over then than with the mass of gold that was equal to it in weight; and thus, from the superfluous quantity of water carried over the brim by the immersion of the crown, more than that displaced by the mass, he found, by calculation, the quantity of silver mixed with the gold, and made manifest the fraud of the manufacturer.
Well, the problem I have with this story is that any air space in the crown, any mixture of light material from a mold, would make the crown take up more volume, without actually meaning any gold was stolen. Short of an analysis actually finding silver in the crown, or at least indicating that the witness who first murmured against the smith did anything but pick up the crown and think "gee, this seems light for its size", I think there should be more than a reasonable doubt that the goldsmith was innocent. Was there ever any proof that the smith did anything but make an imposing-looking crown using all the gold he was given?