AMD's site says the 6850 needs at least 500W. That means you will need an upgraded PSU, so if you don't let getting a new one get in your way then get a nice name-brand one. Ask any PC builder and they will all say never skimp on the power supply. I would recommend a nice Corsair 600W.
A 380W Antec Earthwatts can run a HD6850. The problem is, the "450W" PSU-shaped object in that case is probably not actually good for much more than 300W, and that possibly not cleanly. AMD's 500W recommendation is an attempt to compensate for all the low quality junk out there that is really only good for 60%-75% of what is on the label.
Look for a quality PSU that has at least 22A on its +12V rail(s) in order to have enough juice for a HD6850. More is not harmful, but you don't want less. Anything built by Seasonic will be good. Enermax/LEPA, FSP, Delta, and Super Flower also build good units. Some Corsair are built by Seasonic, but the V2 "Builder" is not; it's made by CWT using some inferior Samxon capacitors known for early failure. I would not call them junk, but capacitor aging will sap their capacity much more quickly than in a better unit.
I just bought a second-hand Radeon HD 6850 1GB GDDR5 and installed into my PC with Cooler Master Extreme Power Plus 460. I know this power supply is at best a 420W, but for my non-gaming purpose the system boots up and runs fine on most applications. However when my kids played the game Sims 3, they complained that the PC is giving noises like "hissing" and "whining" but the game played alright for 3-4 hours stretch. I suspect this is due to insufficient power from the 420W as the HD 6850 required "500W or greater" but then Sims 3 isn't a heavy 3D game. Right now I'm doing regular backups using Acronis True Image so that when my kids crashed the PC I can recover my stuff. Just sharing this experience in case some people want to risk using HD 6850 on their old machines with less than 500W. The PC didn't shut down or crash yet, but there is "hissing" and "whining" sounds coming out from somewhere when playing a simple game like Sims 3.