No, not necessarily. Depends on the age of the board. Some 945 boards won't even run a C2D, stopping at Pentium 4. Just for trivia's sake, did you know that the latest Pentium 4 "Extreme" editions had 1066Mhz FSB?
You can check Intel's website for CPU compatibility. They obviously have the last word. There may be BIOS updates available to accommodate some C2Ds, but I doubt that these would include any 45Nm chips. More likely E4X00 series Allendales would be the most recent useable CPUs.
My best guess says E5000, E7000, and E8000 series are out. (all 45nm)