Parts List:
AMD A6-3670
4GB Kingston RAM (Note that RAM can be had for cheaper, but since it's being OCed, a bit extra spend doesn't hurt.)
MSI Socket FM1 Motherboard (I was able to get a Gigabyte board for cheaper for this particular build, but generally MSI has better build quality, and this is one of the cheapest you can get online)
Samsung Spinpoint 320GB HDD
Logisys Case with build in power supply (We are only running a 100w APU, no graphics card, so this shouldn't be an issue.)
Cost:$259.95
Benchmarking Setup:

To get reliable framerate numbers, I took the flightpath from Moonglade to Schnottz's Landing and back. This takes me the maximum distance, and passes through ever terrain type, including two Cataclysm areas. For these tests I was using the 64 bit WoW, version 5.0.4, Mists of Pandaria. Settings used was the "Good" preset with SSAO turned off.
Frame Rates:

Testing started with the stock CPU clock of 2.7 Ghz, stock GPU clock of 444 Mhz, and RAM at 1333. Moving the CPU clock rate up to 3.2Ghz, which was easily stable with the stock cooler, netted a large increase in the minimum and average frame rates, with very little difference to the maximum. From here, increasing the GPU clock to 825 Mhz, the highest stable rate I could get, average, and especially max frame rate shoots up, while minimum stays the same. This leads me to believe that whatever is causing the minimum FPS, which is at the Uldum river, is because of the CPU aspect, and not the GPU.
From here, I moved up in RAM clock rates. First to 1600, and then to 1866. Both netted an overall increase in FPS. I could not clock higher than 1866 with a stable boot, but the motherboard I used supports RAM frequencies up to 2400. I believe that with RAM speed increase, FPS gain would continue further.
Frame Rates, further exploration:

It was suggested that I use this setup to try LFR after the original posting, so I did, along with a few other graphic settings. Hardware wise, everything is at the previous highest settings, 3.2Ghz on the CPU, 825Mhz on the GPU, and 1866Mhz on the RAM. LFR numbers were taken in the Fall of Deathwing, and are an average of numbers on all 4 bosses. I used the "Good" preset at 1920x1080, and the new SSAO feature turned off. There was no lag during the fights, and though the frame rates dipped down to 22FPS at one point, it did not hinder my play, and was barely noticeable when it happened. I believe that someone could easily do 10 and 25 man raids on this setup without a problem, especially if they turn shadows on low as well, as I'm about to explain.
Knowing that shadows cause the biggest drop in FPS, I went back to the flightpath test again, this time with Good, High, and Ultra presets, except for having SSAO off and shadows on low. Using the standard good settings as before, we see that the frame rate never dips below 30, the go to standard for what's considered "playable" and the average FPS going to to nearly 69FPS, which would easily hit the Vsync cap, if I were to turn it on, providing a smooth playing experience. Moving up to "High" settings, we still see a very nice average of 46.5 FPS, and a low of 25 FPS. Again, though some people can see a slight choppiness at this rate, it does not happen enough to diminish gameplay. Ultra drops things further, averaging only 36.6 FPS, but still barely in the "playable" range. As we see from the LFR test, you would struggle in a large raid or battleground with these base numbers, but for general questing and 5 man dungeons, I believe even "Ultra" settings could be viable.9
Conclusion:
For $260 in parts, this is a solid build if World of Warcraft is the main goal. I was able to play other games at easily playable framerates as well. such as Diablo 3 and Team Fortress 2. It's worth noting as well, that the lowest frame rates all happened in the exact same spot in Uldum, and that there was almost constantly a frame rate over 30 once the APU was overclocked, providing a perfectly playable, and smooth gaming environment.