I just saw the HD 6450 doesnt need much Power and no extra connector so probably it will work with your 300 Watt Power Supply. Just buy it first and if its not working buy the Power Supply. If you want to have a better Graphics Card like the HD5670 you will probably need a better Power Supply.

Recap:

If you want to spend 50 Bucks and dont deal with Power Supply just buy a HD 6450

If you want to spend 105 Bucks to have a twice-trice as strong Graphics Card and a Power Supply that will allow you to upgrade your system in the future and have room for improvments. Buy the Supply mentioned above and this Card:
http://www.amazon.com/XFX-Radeon-DisplayPort-PCI-Express-HD567XZNF3/dp/B0034JLTNG/ref=sr_1_5?ie=UTF8&s=electronics&qid=1314686433&sr=8-5

Just to clarify, is that processor a single-core?

yes its a single core but nevertheless he has onboard now without dedicated vram and that seriously acts like a 10 ton weight on the computer performance. With a 6450 he could run oblivion and mass effect, bioshock but now he cant even run Half Life 2.

Also I know that with a 5670 the CPU will be a bottle neck but he can easily upgrade mainboard and cpu later for a few bucks and keep Power Supply and Graphics card.

It has been awhile since I could take a good look at this.  Thanks for all the help.

It is actually a dual core processor, by the way.

It seems that a 400 W power supply is recommended for the HD 6450 according to AMD's website.  However, the calculator at Newegg only adds 10 W to their recommendation when you add the 6450.  I think I will just try using the 6450 without upgrading the power supply.

Is there anything to watch for to see if it is causing problems?  Or is it going to be obvious if it is not working right?

6450 will draw close to 150W under load. If you are over the limit you will notice it. Symptoms can be varied but can include, System lockups, System crashes/bluescreens, artifacting, program crashes and other random behaviour or even a complete failure to boot.

Do you think my system can handle it without upgrading the power source?

EDIT: I really would rather not upgrade the power source, since I don't plan on upgrading anything else in this computer.  If I need to upgrade again, I plan on just buying a whole new computer.

honestly I think you will experience problems. You might be able to get the machine running with it but I would expect it to be a bit flaky, especially playing games. you really should have a 400W supply for that combo, power supplies are dirt cheap, especailly from many PC stores that have loads of ones they have pulled out of cases for people that want beefier ones. I bought a 500W supply from my local PC store for $20 a few weeks ago, second hand but hardly matters for the machine I put it in, if you ask many times they are happy just to offload some of the lower end spares.

Thanks!

I take back what I said, I didn't realise the 6450 was a very low power usage card. even under load it should comfortably stay within your psu limits. apologies for misleading you.