If you bought the new computer with Professional installed it cannot be moved to another machine, but it can be upgraded to your Ultimate retail copy.
MS wants us to use Anytime Upgrade to move between versions of Windows 7, where you buy a key and unlock all versions. There are some reports that users have been able to insert their retail key into the Anytime Upgrade to get it to work for unlocking to the greater version.
If not, you will have to do a clean install of Ultimate over Professional. When you boot the Ultimate DVD to do a clean formatted install, it will scan the HD to see any prior OS which will pass a flag to allow Upgrade version key, even if you go ahead and use Custom install Drive Options to delete, re-partition and format. If you have full retail Ultimate you needn't worry about that.
There is a workaround to do an in-place Upgrade from one Windows 7 version to another by changing two registry keys to trick the installer into thinking it is doing a same-version Repair Install, which is always allowed. The workaround showing the reg keys to change is here: Upgrade the Windows 7 RC to any retail version « Icrontic Tech
Be sure to set a restore point first.