Will the Radeon 6670 fit in my pc?

I have a Gateway DX4860-UB33P and I'm wondering if the Radeon 6670 would fit in it. http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Im planing to upgrade my power supply to a 430W http://www.newegg.com/Product/Product.aspx?Item=N82E168...

If it wont fit then I have this case as a backup http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Specs for the gateway
Intel core i5 2320 3.00 ghtz
1TB HHD
8GB DDR3 ram

If the motherboard wont support it then I was planning to get this one http://www.newegg.com/Product/Product.aspx?Item=N82E168...

I am worried that the motherboard wont fit in the case or the GPU wont fit on the mobo. Any suggestions,help,tips would be great! thanks!