I did this successfulyl recently. To be real honest, if you don't have a fairly proficient knowledge in Linux, you're in for a hard road.
But I'll throw you the best bone I can.
BACKUP BACKUP BACKUP NOW BEFORE YOU GO ANY FURTHER, YOU BEEN FREAKIN WARNED.

had to get that out of the way.
My /myth partition is XFS which supports online expansion. EXT3 does as well, I'll assume you're using ext3 and this it a mythtv box you're trying to upgrade.
I'm assuming that the partition that you want to grow is the last one (i.e /dev/hda3).
do a fdisk -l as root and make sure, here's mine for example.
dev/sda1 * 1 7 56227 83 Linux
/dev/sda2 8 142 1084387+ 82 Linux swap / Solaris
/dev/sda3 143 38913 311428057+ 83 Linux
Notice how sda3 is the last partition and is the big boy?
What I did:
1. You'll need to use a Linux boot CD and do the copy with both drives unmounted, I used Knoppix 5.0.
2. Once in you boot cd, as root, do this:
type mount and press enter, do you see any of your /dev/hdX or /dev/sdX partitions mounted as read only? if you do unmount them all with a:
umount /dev/sdX
or
umount /dev/hdX (X being the partition number)
Now that it's unmounted its time to clone the hda drive to the sda drive, as root:
dd if=/dev/hda of=/dev/sda conv=noerror,sync bs=4k
be prepeared to wait a while, I copied a 160 Gig to a 320 Gig and it took right at 55 minutes. Make sure you tune your drives with hdparm and enable DMA or you'll be waiting half the night..
3. when finished, reboot and REMOVE your /dev/hda drive as it's no longer needed. The hda drive is completely intact, keep it safe as it's your fallback.
4. boot knoppix again and become root at the command line
5. use parted to expand the ext3 partition, reboot
6. boot knoppix again and become root at command line
7. make a mount point for your mythtv data partition, the one with /usr /bin /sbin on it , it should be /dev/sda1 now. If knoppmyth already mounted it, unmount it as it's mounted read only, you dont want that.
umount /dev/sda1
mkdir /mnt/sda1 (if one isnt already there)
mount /dev/sda1 /mnt/sda1
cd /mnt/sda1
do an ls -la, you should be in the root of you mythtv os partiton.
do a "chroot ."
now edit your /etc/fstab, your /etc/mtab (jusr for kicks) and your /etc/lilo.conf to reflect your new sdaX instead of your hdaX.
rerun lilo: lilo -b /dev/sda
sync your disks: sync
exit out of your chroot: exit
go back to the knoopix root and unmount the SDA drive:
cd /
umount /mnt/sda1
reboot and you should be good to go.
Or, you coudl do what snet suggested.
