Need A Gaming Computer for Under $600

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: AMD Phenom II X4 965 Black 3.4GHz Quad-Core Processor ($90.98 @ SuperBiiz)
Motherboard: MSI 970A-G46 ATX AM3+ Motherboard ($76.99 @ NCIX US)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory ($56.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ Outlet PC)
Video Card: MSI GeForce GTX 650 Ti Boost 1GB Video Card ($109.99 @ Newegg)
Case: NZXT Source 210 Elite (Black) ATX Mid Tower Case ($49.99 @ Amazon)
Power Supply: XFX 650W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply ($49.99 @ Newegg)
Optical Drive: LG GH24NS95 DVD/CD Writer ($14.94 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($89.98 @ Outlet PC)
Total: $599.83
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-07-27 14:46 EDT-0400)

How well can this run BF3?

It should be able to run decently. Don't expect it to max it out, but I would assume it can play at about high.