You can easily get a decent graphics card like that in budget, just need to save a bit on the other parts.
CPU - Games won't even touch the 5th and 6th core, I would get the FX-4100 or FX-4170 instead if you want to stay with AMD.
RAM - I would go with 4GB, games aren't using any more than that yet. You can always upgrade it if you feel the need as well.
If you want to go with AMD, this would be better:
Crucial Ballistix Sport 4GB (2 x 2GB) 1600Mhz
ASRock 970 EXTREME3 AM3+
MSI Radeon HD 7770 1GB
Samsung Spinpoint F3 1TB (Great deal on this at the moment.)
Corsair CX-430 430W
NZXT Source 210
Samsung DVD Burner
Honestly I would change a couple of things. I am going through a rebuild myself. Here are some options you may want to make:
Power Suppy: OCZ600MXSP OCZ 600W Modular 49.99 after rebate
CPU: Phenom II X4 965 Black Edition $109 (Can OC using a 18.5 multiplyer)
MB: GIGABYTE GA-990XA-UD3 $129 (money saved get a better MB & Video Card) also you can add another H679...and xfire at 2x16 instead of having 1 PCI-E @16 and the other @x4
Video Card: H679QNT1G2M (HIS 6970 1GB 256 Bit Memory Interface) faster Core and more shader processors than a 7770 and its Memory Clock is only 25 MHZ $144
Hard Drive: Same
Case w/DVD drive combo'd @ 70.98 (CoolerMaster HAF 912)
Memory: GSkill 8GB (2x4GB) 1333 DDR3
All parts came from Newegg
Your updated budget was 600-625 (met after rebates)