Yes, it is fixable... if you're either a. willing to take some risks, or b. spend loads of money.
If you're going for choice A, you can use a tool (such as a heat gun) to attempt to reflow the solder of the video chip's BGA package. It's not too difficult to do, but there are definite risks.
First, you could run the gun too hot and blow something, or melt the solder too much so that it becomes displaced and shorts things out. I've done both
(both on 14" 800MHz G3 models). Of course, I've also resurrected three of them, though one I've had to do more than once. So yeah, definitely a risky operation.Second, you could screw something up during the disassembly/reassembly process. I've never had a problem with that, but some people without several years of experience (or with a lack of patience or improper tools) can potentially have some problems.
For choice B, you could send the malfunctioning computer off to either an Apple service centre for a logic board replacement, or to a specialty centre that would be able to professionally reflow the BGA on the current board. The former option is generally more expensive than the latter, and many times a replacement board from Apple will exhibit the same failure down the line (I've bought iBooks that were on their fifth logic board!).
Some specialty centres may make you send them just the bare board to reflow (so they can't be held responsible for damage from, and avoid the hassle of, taking the computer apart), so you may still run into assembly issues if you're not comfortable with the process.