Get at least a 400W PSU with a 6-pin PCI Express power connector for the graphics card. You'll probably want something really stable, like this:
http://www.newegg.com/Product/Product.aspx?Item=N82E16817371016That should give you all the peace of mind that you need, plus some spare power. If you can't afford it at the moment, just wait. Cheap PSU's aren't worth the trouble.
In the meantime, you can try installing the graphics card, but check the manual and see if the version you ordered needs the 6-pin power connector to run, and then see if you're current PSU has that. If you plug it all in and it works, then great, you can use it for now, but the strain may eventually kill some components in the long run.
Also, I don't see what's so laughable about that machine. Then again, I've got a 433Mhz Celeron w/192 MB of SDRAM sitting in the corner along with a 133Mhz Pentium desktop.