http://www.fiatbarchetta.com/
Ah. Yes, that must be it, but somehow it doesn't look as good as I remembered it...
Well, for whatever it's worth, the reason I was so confused is that despite having an Italian name, the thing I was thinking of isn't a Fiat at all. It's a Suzuki, it's called the Cappuccino (more), my brother's fiancee's sister's husband has one, and I got to ride in it (briefly) last weekend. It rules.

Peter