Will an ungrounded electrical plug designed for France work in an electrical socket in Italy?

It depends on the thickness of the pins. The "small" Italian sockets have holes that are the same distance apart as the French/German standard but they're quite narrow so you can't fit a German "Schuko" plug into an Italian socket even though it looks like you should be able to. So if the French pins are like the German ones, no.

(You can however force them into the "universal" small/large Italian sockets to the detriment of the latter!)

On second thoughts: try plugging it into a bathroom shaver socket! They seem to be universal across Europe - even in the UK with its ridiculous plugs. If it fits a shaver socket, it'll fit in Italy.

You can always buy a German-Italian adaptor in any Italian supermarket as a fallback.

Take a look at this:

http://www.interpower.com/icl/guide.htm

If it's a "B" then it should work. Not if it is a "F".