You are correct - putting an existing Windows installation into different hardware (ie a different computer) won't work, Windows will just refuse to start. You have to re-install Windows on that hard drive after you've installed the drive into your new build.
Boot from the Windows 7 DVD and it will show the existing partition that's got Windows already installed on it.
Select that partition and "Delete" it. That will then show as "Unallocated Space".
Select that "Unallocated Space" and install Windows on it.
Windows Setup will automatically create a new partition there, it will then format that partition, then it will proceed to install Windows on it.