If you have 20GB of free space on the drive, just use a LIVE CD/USB and gparted to "copy and paste" the existing Ubuntu partition to that area.
Originally Posted by IqAndreas
Change the new partition's UUID (search for detailed instructions), then reboot into Ubuntu and do:
Mount the new partition and change the /etc/fstab file in it to use the new UUID.
Reboot into the new 20GB Ubuntu system, then:
And install the boot loader.
sudo dpkg-reconfigure grub-pc
Now you will be booting Grub off the new Ubuntu partition and you can now delete the old 10GB partition.