On amd's site 400w psu is recommended for that card. But yet again it's just the safe recommendation.
Did you try the psu calculator that I linked? It should give you the theoretical minimum wattage output recommendation for your specs. So if you don't have your cpu overclocked, multiple usb devices, fans, hdds you should be ok. But as Shane said using psu its max output will decrease its lifespan, which could lead to a broken psu and in the worst scenario broken psu could broke the other pc parts.
According to the PSU calculator power need difference for the nvidia geforce 210 and for the hd 6670 is around 30watts. Is this ~30w increase too much or not, I can't say. You can only take the risk and test it. I think many shops will accept returns if you say my psu couldn't handle it.
Edit: And don't pay for the 30$ extra job for them to let them put it in, it's 50% of that card's value! Unless you really want to have the advantage to plain them if it doesn't work!
Edit2: And you could get quite good new psu for that money!