The tpu link should show you what you need to do but the important parts are -
Make sure the BIOS switch on the card is set to the 1 position (pictured further down if you don't know where it is).
Run ATI Winflash, click Save to save your BIOS, so you have a backup in case something goes wrong.
Load the HD 6970 BIOS into Winflash by clicking Load Image, followed by Program.
Let the flashing process complete and reboot your system.
If the card doesn't boot you just have to switch the bios back to the other setting and it will be fine. After that you would just reflash with the bios you saved (to the unworking switch position - not the one you booted with!) That's all it takes basically, you won't go wrong because you should only ever write over ONE of the bios's meaning you can always just switch to the other to recover it if it doesn't work.