I'm just slightly over budget on this, at ~$737 after shipping, but cutting anymore corners would kinda suck.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814130826
GF GTX 660 Video Card
http://www.newegg.com/Product/Product.aspx?Item=N82E16817151096
Seasonic 620W PSU (best warranty and gets the best reviews, could probably pick another manufacturer if you really needed to save a few bucks)
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231636
1 stick of 8GB DDR3-1600. Ironically cheaper than 2 sticks of 4GB. Not ideal but we're on a budget, ya? :p
http://www.newegg.com/Product/Product.aspx?Item=N82E16813157303
ASRock Intel H77 chipset mobo. Just fine if you're not overclocking.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116782
Core i5 Ivy Bridge. Note that it lacks the video portion, so make sure your video card is working.
http://www.newegg.com/Product/Product.aspx?Item=N82E16822236339
WD Blue 1TB drive. About as cheap as you're gonna get without buying the awful WD Green.
http://www.newegg.com/Product/Product.aspx?Item=N82E16832416550
Windows 8 64-bit. Not a huge fan of Win8, but buying Win7 now is like shooting yourself in the foot. Not to mention, Microsoft will be releasing a free update for it this year that adds back much of the missing Windows 7 stuff (like the START BUTTON). They're also the same price.
This build obviously assumes you'd be bringing your DVD drive and case over to be reused. You probably can't reuse the RAM because everything runs on DDR3 now, and back then almost everything was DDR2.
Comments on this build:
A. It's going to last a while, it's not bargin-basement parts.
B. The video card especially is the cheapest you can get without going into budget territory. It'll play everything on high settings right now.
C. You can save some cash with a Core i3 CPU as well if you want, quad-core still isn't THAT important. But the price difference is so marginal you might as well get an i5.
D. The biggest thing missing is an SSD, but at $700 it's just not realistic. SSDs are amazing though, so it should be considered as the next upgrade for this PC. It just makes the whole system far more responsive.