Yes! You will need a new PSU. Anything from Corsair, Antec, PC Power and Cooling, XFX, or Seasonic would work just fine. I would get at least 500w or 650w if you plan on CFing on the 12v rails. OCZ PSUs (new ones) are pretty good too.
The power draw difference between those cards is a theoretical max of 36W and will likely never even be that large. Is that AcBel the best PSU on the street? No. Is it likely sufficient for your minimal upgrade? Yes. Give it a try. If you're really unsure, then call Asus and ask their tech support....especially if the PC is still under any type of warranty.
It looks like the maximum combined +12 Volt continuous current rating is 18 Amps.
A Radeon HD 6670 can draw up to 5 Amps from the +12V rail. That leaves you with 13 Amps remaining on the +12V rails to power the CPU, motherboard, HDD, ODD, CPU cooling fan, case cooling fan, which should be sufficient as long as you don't start adding more devices.