You probably don't need a large power supply, but it all depends on the type and amount of hardware you have on the system. If you have 4 hard drives, and every PCI slot is being taken up, then of course, it probably won't work. However, if you have a standard configuration with one or two drives, maybe an extra network card -- should work. Look at the processor. i5 and i7 processors can work on power supplies that were technically designed for Pentium 4's due to their low consumption on idle.
I would say you have a fair chance. To reduce power consumption on a Micro-ATX machine with a 450W supply, I replaced 2 HDs with SSD's. While this is not the cost effective solution, it runs fairly well despite the few crashes when running at 100% cpu non-stop for over a week. This was probably due to some kind of calculation error with the CPU. I suspect. One thing you should worry about is if you have enough of the right connectors for the card since it now uses 2 to power the card. You will need an additional molex connector adapter for the right set of prongs.
Assume it doesn't work: your waste case scenario would be to buy another PSU, really.
If you go here: AMD Certified Power Supplies you will see a list of ATI certified PSU's. It will at least give you an idea of what you will need depending on which card you want to get.. nVidia should also have a list of certified PSU's on their site though I'm not 100% sure on that as I've always been an ATI guy..
Lorenkjr, according to AMD only a 500w PSU is needed for a 5870. This is misleading whilst being true. The 500w PSU needs to be kicking out 40amps between the 12v rails otherwise problems may be encountered.
Try some of the reviews found at Guru 3D.com as they always include ampage which is more important than watts.