It would be close and not within my safety bracket, but yes, i believe it will work.

You may want to still consider a new Power supply. The AMD(not ATI) HD6670 is not a hardcore gamer card.

It will run most games but what are you planning to play at what resolution?
 
how would have i known a 320psu means 320w? hardly anyone would do so

anyways, you need more wattage. i suggest a 500w to get a nice gaming card like a asus 7850 (if it fits) or a msi 7850.

for the psu, i recommend the pc power and cooling mk3 500w. usually sells for 50-60 dollars