Firstly, it IS possible, but you will have to make some compromises. One of which is probably reducing your resolution when playing a game to 1600x900. Honestly, while not what you want, its not that huge of deal, and you will get better frame rates with a 6850. Also, you will probably have to forgo AM3+ upgradability. If you are ok with that fine, if not, further compromises will have to be made (probably in the processor and/or GPU). Here is a build I put together that is roughly in your price range. You could save some money by switching the PSU I used to an Antec 400 NEO (but this is more expensive after the XFX MIR, but you cannot count on the MIR, just hope)
LITE-ON 24X DVD Writer Black SATA Model iHAS-324-98B
Item #: N82E16827106334
Return Policy: Standard Return Policy
$19.99
NZXT Source 210 S210-001 Black “Aluminum Brush / Plastic” ATX Mid Tower Computer Case
Item #: N82E16811146075
Return Policy: Standard Return Policy
$39.99
Recertified: Western Digital Caviar Blue RFHWD1600AAJS 160GB 7200 RPM SATA 3.0Gb/s 3.5" Internal Hard Drive -Bare Drive
Item #: N82E16822136590
Return Policy: Limited Replacement Only Return Policy
$66.99
XFX ProSeries P1-450S-X2B9 450W ATX12V / EPS12V SLI Ready CrossFire Ready 80 PLUS BRONZE Certified Active PFC Core Edition ...
Item #: N82E16817207018
Return Policy: Standard Return Policy
$58.99 (not incl $20.00 Mail-in Rebate)
ADATA XPG Gaming Series 4GB (2 x 2GB) 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) Desktop Memory Model AX3U1600GB2G9-2G
Item #: N82E16820211409
Return Policy: Memory Standard Return Policy
$29.99
AMD Phenom II X4 955 Black Edition Deneb 3.2GHz Socket AM3 125W Quad-Core Processor HDZ955FBGMBOX
Item #: N82E16819103808
Return Policy: CPU Replacement Only Return Policy
$124.99
Microsoft Windows 7 Home Premium SP1 64-bit - OEM
Item #: N82E16832116986
Return Policy: Software Standard Return Policy
$99.99
COMBO
SAPPHIRE 100315L Radeon HD 6850 1GB 256-bit GDDR5 PCI Express 2.1 x16 HDCP Ready CrossFireX Support Video Card with Eyefinity
Item #: N82E16814102908
Return Policy: VGA Standard Return Policy
BIOSTAR A770E3 AM3 AMD 770 ATX AMD Motherboard
Item #: N82E16813138179
Return Policy: Standard Return Policy
$184.98 (after -$15 combo and not incl $15 MIR)
1
Subtotal: $625.91
Shipping: $11.60
Grand Total: $637.51