Boot a live CD with both HDDs attached, start the Partition Editor (gparted), copy and paste the partitions from one drive to the other:

1. In the top right corner select your old hard disk.



2. Select a partition you want to copy (right-click → "Copy"):

   
   
   

   




3. In the top right corner select your new hard disk.




4. Paste the copied partition into unallocated space (right-click → "Paste"):

   
   
   

   




5. Resize if you want (right-click on the partition → "Resize/Move")

This will keep the unique identifiers of the partitions, so you shouldn't need to adjust anything after cloning the drive.

The final part is to re-install GRUB:

1. Note the root partition name of your new HDD (for example /dev/sdb1).



2. Open a gnome-terminal and run (sorry, I don't know a GUI way to do it, but it's just a couple of commands):

   
   
   
   
   
   1. sudo mount /dev/sdb1 /mnt
   
   
   2. sudo grub-install --root-directory=/mnt /dev/sdb
   
   
   3. sudo umount /mnt
   
   
   
   
   

   Make sure you use the disk name at the end of point 2!

That's it, reboot with only your new HDD and you should be fine.

Some advise: Do backups first! Only run one operation at a time and not all at once (so: copy → (resize) → copy → ...).