Curiously, IIRC, natural born American citizens can't hold dual citizenship??? I don't know where I got that from, but it seems that I read somewhere long ago.
No such law, never has been. It's certainly not encouraged or advertised but there's nothing in US law than prevents dual citizenship - it would be very hard to enforce anyway, though there are laws/rules that makes your 'other' citizenship basically meaningless (travel/tax etc) unless you choose to go back and live there.