Carolus Rex - I guess I would have accepted your surrender to my allies the barbs
Sten Sture - my experience has been that any barbs adjacent to your city will disappear when you pay them off. Any barbs not adjacent to that city keep looking for trouble.
With respect to paying off the barbs: I have never had it fail in single player on v2.42. Imagine my surprise in one of my early MP games when the barbs walked right in without demanding money!
I disagree with those who advocate letting the city fall before you have a diplomat built. That city will start churning out 24-shield archers, which will then threaten other cities. It's not pleasant to watch the domino theory at work on your own civ!