Will it work?

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=N82E16814102870
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.