I thought you had a 600W PSU?
I have a PSU already, 600W...
The Dynex 400W is actually a rebranded Hunkey Green 450W and their circuitry is identical and at the same time rubbish.
You will need a new PSU regardless of your decision which does make it difficult. Any graphics card you install (AGP or PCIe) will need a better PSU, period. Why?
Due to its old ATX design standards, that PSU dedicates 12V+2 (15A) for the CPU only. That leaves only 12V+1 (14A) to the rest of the 12V system. In theory that is 168W max. At this point the PSU either shuts down of explodes (as does its exact cousin the Huntkey 450W).
Now given the TDP of a HD4650 (RV730) is 75W or less (6.25A) you have only around 75W to keep the rest of your system running happily, thats at MAX.
Essentially for any discrete graphics card I don't recommend anything other than a total of 20A available. This is minimum.
So as you can see a new PSU is needed (see EDIT
* below).
So, to answer your question you can get:
PSU
*: $30 ($6 shipping) Cooler Master 400W that can provide 23A on the 12V rail (not great but much better)
http://www.newegg.com/Product/Product.aspx?Item=N82E16817171042&cm_re=400W-_-17-171-042-_-Product
Motherboard: $70 (Free shipping) - Biostar H67 motherboard which supports Core i3, i5 & i7
http://www.newegg.com/Product/Product.aspx?Item=N82E16813138310&cm_re=h67-_-13-138-310-_-Product
RAM: $45 (Free shipping) - 2GB DDR3
http://www.newegg.com/Product/Product.aspx?Item=N82E16820148243&cm_re=ddr_3-_-20-148-243-_-Product
CPU: $125 (Free shipping) - Core i3 2100-
http://www.newegg.com/Product/Produ...078&cm_re=core_i3_2100-_-19-115-078-_-Product
GPU: $65 (Free shipping) - HD5570 PCIe -
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150485
I know the total is above $250 being $340, but you get a new PSU as well
*. Something that is needed regardless.
The above items are for illustration only, but you can see (without the PSU) you are only spending $60 more than your budget and getting a much, much, much better and faster system. One that will take you into the next 5 years with upgrade room (you can go to Core i7 (LGA1155) and any PCIe graphics card provided you have a sufficiently powered PSU). Btw the HD5570 will belt anything you are considering now.
Sell you current mobo, cpu and ram and get may be $50? That makes it net $256 ($306 worst case) in real terms. But now you have a fast, up to date massively more powerful computer, and you haven't wasted $125 on a AGP graphics card.
Re-use your current hard drives, case etc.
*edit* you have changed your post so you have a 600W PSU afterall, please post specs, but assuming you can use that, you now have the above computer for $270 without selling anything. Not bad i reckon. Totally worth the extra $150 over the cost of the AGP card.