Assuming it's actually the logic board that is the problem, you can probably get a replacement board for $200, and it would then take you a few hours to replace it, if all goes well. If you are not up for an advanced technical challenge, such as this would be, you can buy a fully working equivalent machine on eBay in the range of $350. The choice, of course, would be up to you, and whether you want to take on a technical project, or just bite the bullet and acquire a working computer as soon as possible.
But as Matthew mentioned, let us know what symptoms you are seeing, and perhaps the problem is not as severe as they told you at the Apple store. For instance, does it make the chime sound when you power it on? Is there anything on the screen, and if so, what? What color is the light on the power plug?