I don't have a 1DIII but I have a 1DsII. And the 1DsII kills the 5D in terms of AF speed and accuracy. The 1DIII will only be better.
In a very dark reception hall, my 5d beats my 1dm2 easily. Obviously I am using flash, and it could be a factor of the AF assist not reaching far enough.
Anyway, I'll be looking for a used 1d3 within the next couple of months.
1dm3


Although, it's so rare with the 1DsII.
