Ok, so I summed up the past suggestions in the following build:
: Intel Core 2 Duo E6300 1.86Ghz
: Gigabyte GA-945-S3Intel 945P
: Kingston 1GB DDR2-667
: eVGA GeForce 7900GS 256MB
: Sparkle 400W (36A on two +12V rails)
: Western Digital 250GB SATA 3.0Gb/s
: LG 18X Super-Multi DVD+/-RW
: Xion Solaris XON-406 Black
Slightly over-budget, but not too much. The DVD burner is not crucial, so one could save $30 by removing it.
As you can see, I focused mainly on the video card, since it's the one part that can make a large difference in games. A 7900GS has a significant lead over a 7600GT.
I choose a single 1GB module instead of two 512MB ones because this system is meant to be upgraded with a second GB of RAM soon after being built. It's better to end up having 2x1GB than 4x512MB (RAM slot overpopulation is rumored to affect latency).
I also had to select a cheaper case, but it's easily switchable for any other case with a similar price.