I wouldn't be nervous running it on solid 300W, but I would with a cheap 300W. It will most likely be fine since the 5670 uses so little power, I just will always buy a solid PSU because of experiencing the hundreds of $$ the blowing of a cheapo PSU can cause (even when the PSU is rated way over what you should actually need).
Don't mind me though- I'm just a PSU snob. you should be fine:-)
Snob as it may be, I completely agree. I suggest getting a decent 400-500W PSU first. From there, you can expand your choices for a new graphics card. But if money is tight, HD5670 it is.