Assuming the hard drive containing the clone is installed and mounted in the new MacPro...
Booting to an OS X installation disk (which includes the recovery partition installed with Lion), you can copy the the contents of the disk image from CCC onto your new drive by selecting .dmg file creating by CCC as the source and your hard disk as the target. Unless some drivers are missing out of the box (which is quite possible if your MacPro isn't currently running an up-to-date version of Lion), after the copying process is complete, you can reboot your new computer and it should be as though you never left your old one.
Alternately, you can use the Migration Assistant from the installation disk (or in the /Applications/Utilities folder). From the wizard, select "From another Mac, PC, Time Machine backup, or other disk" and then hit next and select "From a Time Machine backup or other disk". Select your CCC .dmg file and presto!