Upgrading my GPU to Sapphire Radeon 7850

[Akersar]

So yesterday, I bought the Sapphire Radeon 7850 for $166 without tax lol. I was wondering if my XPS 8100 would be able to handle it. I researched for about 7 hours (not straight xD) and it seems safe to say that I will be able to run it with my 350W PSU. It seems as though Dell underrates their PSU quite heavily.
My PC Specs:
i5-750 @ 2.67 GHz
6 GB of RAM
Current graphics card: Radeon 5670 Here is the exact one I bought: http://secure1.ncix.com/products/?sku=77258&promoid=1016
PSU: 350W Dell standard.

I need your opinions!!! Thank you!
EDIT: If you need me to buy a PSU, please suggest a good quality one that can support the rest of my specs that is around $40. I live in Canada.

 

[Onus]

Based on the PSU sizing calculator at http://www.extreme.outervision.com/PSUEngine and using my arbitrary assumptions about drives and fans, plus allowing for 10% capacitor aging, a 337W PSU is needed. If I increase that to 25% aging, 376W is needed. So if your machine hasn't been ridden too hard, it ought to work, but you may be close to the limit. When you can, a PSU replacement might be appropriate, just to be safe.

 

[Akersar]

The only worry I have is that the GPU requirement states it needs 500W PSU, and I do not really want to spend more money at the moment. And is it possible to harm my computer if it doesn't work?