Will the Logitech Bluetooth Keyboard Case "for ipad2" work with the new ipad? I have an ipad2, and I have this keyboard case: http://www.logitech.com/en-us/1222/8213

Will the "new ipad" fit nicely into this case, physically?
Or, if I get one of the new ipads, will I need to get a keyboard case specifically "for new ipad"?
Yes, it fits. I bought the iPad 2 case and it fits my third gen iPad...$20 cheaper too :)

The latest generation iPad is slightly thicker then the iPad 2.

The iPad "2" is 0.34" (8.6 mm) depth
The iPad "3" is 0.37" (9.4 mm) depth

Otherwise they are precisely the same width/height.

While I don't have the case/keyboard you mentioned, it looks like it may fit in there. If the case has a solid nook for the iPad to fit into, it likely will get (noticably) tighter - but it entirely depends on how much gap is provided. Obviously since the new iPad hasn't been physically sold yet, there wouldn't be a way to actually test - other then measure with something like a micrometer.