I'd follow the guidelines in my signature. We need more information, especially if you need to include an OS within the $300 budget. If you need the OS included, it's impossible. Windows 7 will run $100, which means you've got $200 for the tower, which literally cannot be done.
Personally, I don't think $300 is going to cut it. The bare minimum that you could spend and get a decent machine is around $400, and that's just with the CPU, motherboard, RAM, GPU, optical drive, and HDD. Off the top of my head, a CPU is going to run about $70, the board about $60, the HDD is going to be $40 (the 7200.12 500 GB), RAM is $25 (Kingston's Hyper X 2x2 GB kit), the DVD drive is under $20, the GPU is going to be $100 (the HD 5770), and the case and PSU could be had for around $80. That's including rebates and ignoring shipping. I'll throw a couple of build together below as examples.
You could build a system that didn't use a discrete GPU, but it wouldn't play WoW very well. Even then, I'd want an i3/H55 system at the minimum, and that would be about $225, which is basically the cost of the cheap CPU, board and HD 5770.
I would wait the couple of months just to increase the budget a little. By then, Intel's new Sandy Bridge CPUs will have been re-released (there was a motherboard design flaw that forced a recall), which should drive prices down.
CPU:
X3 435 $72
Mobo:
ASRock 880GM-LE $60
RAM:
Kingston Hyper X 2x2 GB 1600 mhz CAS Latency 9 $25 afer rebate
GPU:
HD 5770 $100 after rebate
HDD:
Seagate 7200.12 500 GB $40
Case/PSU:
APEX PC-389-C and OCZ StealthXStream II 500W $55 after rebate
Case fans: That case doesn't include any, so I'd pick up a couple. 1x
80mm for the front, 1x
92mm for the back. Total $8. You could get by with only the rear fan, saving a couple of bucks.
Optical:
Cheapest SATA DVD burner $17
Total: $377, not counting shipping or taxes.
You could drop out the HD 5770 and get really bad performance, as the board as onboard graphics. That's fine for watching movies and such, but it's horrible for gaming.
EDIT: Forgot about the
GTS 430 being a viable current generation option. That would drop the total price $50.