Will the sapphire hd radeon 6670 fit in a dell inspiron 620 mini tower

If you have as you say a mini tower then the card should fit , The only time you would run into problems is if you had a slim case and then you wouls need a low profile card with a low profile bracket. In looking online at the Dell Inspiron 620 it is a mini tower so a regular size bracket should fit. The dimensions of the card that I think you are looking at are 6.46" x 4.21" x 1.57" and these measurements can be found in the details tab along with the power supply requirement of 400w. So the card is 4.21" tall so if you measure the back of the case where the bracket would go and allowing for the bracket to slide into the bracket slot you need about 4" from the motherboard to the top of the bracket rail where the screw hole is to attach the bracket to the case. If you don't have 4" then you need a low profile card and bracket like this one;

http://www.newegg.com/Product/Product.aspx?Item=N82E168...

The dimensions of this card are , 7.48" x 2.44" x 0.91" , so you would need 2.44" of space for the bracket.