Well, the method of Archimedes will still work. You need a graduated cylinder of adequate diameter, callibrated in milliliters or finer. I know the coin has displaced so many ml of water. Now the book says the density of pure silver is 10.49 grams per cc. So a nearly pure silver tetradrachm that weighs 17 grams should displace about 1.6 ml, unless I did the math wrong.
Most jewelers only use balances accurate to a tenth of a gram.
I have no idea how analytical laboratories measure the volume of very small objects, but I imagine it's just some refinement of the displacement method.