This is hard to do on a live system. Essentially you need to do the following:

1. copy the files from /var to your new partition: ie cd /var ; find . -print | cpio -pdmua /newvar

2. You then need to edit /etc/fstab to have set you newvar to be mounted as /var when your system boots.

The tricky bit is removing all the old stuff from /var before you can safely reboot. I would suggest you boot of the install CD, then mount the ubuntu parition (may be done automatically. Then

sudo mv /var /oldvar # (on the correct partition.
sudo mkdir /var # so there is an empty partiction to mount your newvar onto.

The after you have edited /etc/fstab you can reboot and all should be well.

Once you are happy with everything you can delete /oldvar