Ok, I've cut all the corners I could (in good conscience) for this build.
Definitely last-generation, but will suffice for the stated purposes. I've had good results with ASRock boards.
ASRock 760GM-GS3 AM3 AMD 760G Micro ATX AMD Motherboard
Model #:760GM-GS3
Item #:N82E16813157215
$54.99 $54.99
Once again, for the stated purposes, this CPU will be sufficient. I don't think going to a single-core is a good idea at all, as varied and complex as online content is getting.
AMD Athlon II X2 250 Regor 3.0GHz Socket AM3 65W Dual-Core Desktop Processor ADX250OCGMBOX
Model #:ADX250OCGMBOX
Item #:N82E16819103903
$58.99 $58.99
A single stick will only run in single channel mode, but you specifically said "cheapest." You can always add another stick if you like.
G.SKILL NS 2GB 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10666) Desktop Memory Model F3-10666CL9S-2GBNS
Model #:F3-10666CL9S-2GBNS
Item #:N82E16820231395
$24.99 $24.99
I am satisfied with the quality of the cheap Rosewill cases, having bought a half dozen different ones and having no problems with any of them.
Rosewill R218-P-BK Black SECC Steel ATX Mid Tower Computer Case
Model #:R218-P-BK
Item #:N82E16811147073
$29.99 $29.99
Buying a cheap PSU is never a good idea. This one lacks a PCIE power connector, but with no expansion plans ever, that won't matter.
SeaSonic SS-300ET Bronze 300W ATX12V V2.3 80 PLUS BRONZE Certified Active PFC Power Supply - OEM
Model #:SS-300ET Bronze
Item #:N82E16817151086
$39.99 $39.99
For performance, the F3 is better, or a WD Black or Seagate 7200.12. Once again, where "cheapest" is the operative word, this choice should suffice.
SAMSUNG Spinpoint F4 HD322GJ/U 320GB 7200 RPM SATA 3.0Gb/s 3.5" Internal Hard Drive -Bare Drive
Model #:HD322GJ/U
Item #:N82E16822152244
$42.99 $42.99
One optical drive is much like another. LG, Lite-On, Asus, Sony; none have performed notably differently or given me any grief.
LG DVD Burner - Bulk Black SATA Model GH24NS50 - OEM
Model #:GH24NS50
Item #:N82E16827136177
$17.99 $17.99
Subtotal: $269.93
Under $300. If you want to spend $300, get the additional RAM stick and a 380W Antec Earthwatts PSU, which would give you the option, if you ever changed your mind, of running a more powerful video card.