Save some more money and get a decent GTX 460. The GTX 560 is essentially an overclocked GTX 460. Currently, I have no problems playing any of the current games out there (granted, my card is overclocked to 900mhz @ the core and 2000mhz @ the memory with temps under 60 degrees C). The GTX 460 is still a very capable card and shouldn't be underestimated. Also, realize that many people here in the forum must have the "best of the best, all of the time", and often downplay the usefulness and capability of previous-gen tech.
As far as my recommendation goes, try not to skimp out on the essential hardware of the build (processor/motherboard/memory). This is how I tackled my recent build (see my sig), and just stuck in a GTX 460. I couldn't have made a better choice. Now, with the money I saved and the money I have been saving since my build, I'll be able to upgrade my video card during the next gen releases.
I think I spend a total of $625-675 for my build (again, see sig), minus the OS and peripherals. You just gotta be patient and find the deals!