I’m totally with Ken Dorney! And if anyone wants an ancient-coin necklace there are plenty of nice looking fakes that can easily be inserted into a screw-together mount, which this one looks to be. (And which will greatly impress the great majority out there who aren’t collectors.)
A quick search on eBay for “silver coin bezel” produced many of these exact bezels. Average price is about $10.00.
Thanks to everyone who gave me advice, especially to those who pointed out how easy it is to remove this sort of mount. I can report that it went off without a hitch, and neither the coin nor the mount received any damage. I'll hang on to the mount in case I think of some other use for it. The edge of the coin looks just fine, and for <$20 I don't think it was a bad buy. Here's a hasty photograph of it: