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.