Yes but with a caveat. It depends on the other components in the system. Also is the PSU working at it's best? The only thing to do is to try it since I assume it's an older one. If it doesn't cut it opt for a new psu from a good oem and good effiency and a higher supply rating, say 550+. Personally I think a 600-700 unit is the sweet spot. PSU work best and last longer if they are not pushed and stressed to the last watt, a little overhead is a good thing. Nothing better than getting something good on sale.
I agree with topper743. It depends on the total components in your computer and the age of the PSU. It might work, but it might not. Only way to find out is to try and see what happens. If you get artifacts or anything else strange, the card may be starving for power. If you stay with a single card, a quality 550wt PSU should work fine. Unless you're running a lot of hardware, there's no need to go larger with a single card.
At the risk of sounding redundant, I have a similar question with a few twists.
1. Did I read this thread correctly? The Radeon HD 3870 will operate safely with no PSU alteration?
2. Is there a better deal anywhere? The card's running about $140 retail right now. Would this be a good enhancement for my system if I wanted to play some higher end games? (Not looking to play Crysis) I'd rather not pay this much, but I'm okay with paying more or less if I'm getting a good value.
3. When I purchased this system I upgraded the GPU slightly from the default setting in order to play WoW smoothly, and it gave me 35-60 FPS, slowing down only in high load areas. Essentially, what I'm wondering, is if this will give me a high frame rate using high graphics settings in current FPS's probably including Left 4 Dead, TF2, Unreal 3.