If I found them, those coins would have been listed on eBay that evening.
In Britain, when a find is historically important, it can be classiffied as "Treasure Trove". That means that the Government has the right to the find but it also means the Government has to pay you full market value.
The process is kept fair for the very reason that, if finders are cheated, they will keep their finds secret to sell secretly at market value instead of notifying the Government.
It's a good deal all around. The public gets to see the finds at a museum and the finder gets his money.