I would unplug the original harddrive from the motherboard (leave it in the case, though) and plug the new one in. Boot from your original Ubuntu install CD, install Ubuntu on the new harddrive so you get all the partitioning and the GRUB boot loader set up correctly. Then switch off the machine, plug the original harddrive back in again and boot from the Ubuntu installer CD to a command-line prompt. Mount the new harddrive's OS partition, erase all the files from that OS partition, mount the original harddrive's OS partition and copy all the files from the original harddrive to the new one. Do the same for any other partitions you had set up. Switch off, unplug the old harddrive, you should now be able to boot from the new drive.
Make sure you don't get confused between the old and new harddrives, obviously. If you're moving from IDE to SATA you might need to change /etc/fstab to mount /dev/sda0 instead of /dev/hda0, or similar.