Will it work?

the 5670 will work on a 300w PSU as long as you did not buy it off ebay for $10, many people run it off DELL, HP, etc., PSU.

Nvidia/ATI over state requirements to cover themselves from people who purchase very cheap PSU.

*BTW, im sure newegg or tigerdirect have the card for much less. even frys or microcenter.

thank you

o and is that card a big upgrade from a ATI Radeon HD 2600 XT?

The HD5670 should be approximately 150% faster than the HD2600 XT. That Best Buy card is waaaaay too expensive though. You would be much better off ordering online. Here is the same card for $40 less and $65 after the mail in rebate.
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
You likely wont have to pay sales tax either making almost $80 less in the end.
It will likely be fine on a 300w but we can't say for certain without the brand/model of the PSU.