Let me step in here and give you guys some info as a person who fixes dead mac logic boards for a living.
1) DO NOT PAY SOMEBODY TO FIX YOUR DEAD ATI CARD, IT IS ALL COMPLETE ********. yes, there is nothign wrong with the BGA on your GPU, the problem is with the GPU itself. the reball is not the reason the card works again thereafter, it only works again because you have heated the GPU up to rework temperature which literally warps the die back into shape.
2) for this same reason, DO NOT PAY SOMEBODY TO REPLACE YOUR GPU ITSELF. there has not been a source for replacement 216-0811000 chips at all. I have tried various sources, as have all the other people in my industry, and the only thing you can find are USED REBALLED GARBAGE being sold as "brand new". they literally polish off the old markings on the GPU so it looks new, then print/engrave fake date codes onto the things, but if you have a good microscope, you can see the signs of this. so with these new chips, when you install them onto a board, they will work again because THE DIE HAS BEEN BROUGHT TO REWORK TEMPERATURE. with laptop chips, you can often get used rebaled chips like this with a bit of life into them, but 100% of these desktop chips have been pulled off of their PCB because THEY DIED. so you will be getting a chip that will work for a few months then take a ****.
3) DO NOT PUT YOUR CHIP IN THE OVEN, OR BLOW IT WITH A HEATGUN AND THINK YOU ARE GOING TO HAVE A PERMANENT SOLUTION. yes it will often bring your card back to life because it warps the die back into shape, but it will fail again if you put the card under any load. you can squeeze a few years out of it if you give it the old "reflow" (tip: if you are doing this with a heatgun, you aren't going anywhere close to rework temperature. at reflow temperature, you can gently poke the chip and it will feel like it is attached to teh board by jello. if it's hard as a rock, which it always will be because there is a reason we use 2000watt rework machines on multilayer PCB, it isn't molten, and you sare actually weakening the solder joints.), and don't put the GPU under real load again, but if you want to game, don't even bother
4) DO NOT BUY REPLACEMENT APPLE CARDS ON EBAY FROM CHINA. they are 100% just "reflowed" garbage. they will last a few weeks/months then take a ****. THERE ARE NO PEOPLE PULLING THESE CARDS FROM WORKING SYSTEMS, BESIDE A HANDFUL OF PEOPLE UPGRADING THEIR CARDS LIKE IN THIS THREAD. it is not worth the gamble.
5) ESPECIALLY DO NOT BUY A 6970. seriously, I feel really sorry for you guys that just "upgraded" a 6770 that may have given a few more years of life to a 6970 that will **** out in a matter of weeks/months under any load. even if the card was a legitimate pull (pro tip: it wasn't, I don't care what the seller said or how well you think it's working, *it wasn't*), it will still cook itself to death sooner than later. your best bet if you DID buy one of these cards is to use something like macsfancontrols to keep the optical drive and HDD fan running at 80% constantly, or even better (and yes, I realize this sounds a bit crazy), cut a window in the back of the mac right above where teh GPU heatsink is. if oyu can keep these cards under 70 degrees under load, they are FINE, but at 80-90, ticking time bomb.
anyhow, I hope I saved some of you guys some wasted time and money with that one, though I'm sure the lot of you are, like me, too stubborn to be told why not to do something until you do it yourself and learn first hand why not to bother doing it ($200 in 216-0811000 later, lol). the reason I am in here is because I'm trying to find an inexpensive, RELIABLE solution to bringing these 2009-2011 imacs back to life so I can turn the pile of dead units I have into cash. I'm not there yet, but I'm scoping out the cheap lineup of quattro cards and thinking an apple geforce flash might work. I'll update this thread as I go, regardless if I succeed or throw in the towel.