Can you confirm that is in fact a slimline PC? (slot for the card in the back of the PC (the bracket) around 2.5 in instead of around 4in) If its a normal sized PC we should be taking you in a slightly different direction (full sized cards run slightly cheaper than the low profile ones).
If it is slim, I agree with the above. If you are interested, here is a slightly more powerful model of the above card:
SAPPHIRE Radeon HD 6670 1GB 128-bit GDDR5 PCI Express 2.1 x16 HDCP Ready Low Profile Video Card (100326LP ) http://www.newegg.com/Product/Product.aspx?Item=N82E16814102958
Should make the call on whether to upgrade with that 4670, the 6570, or the 6670 based on how old the PC is and how much the performance matters. Any of them is a jump over an old 8600, however the higher model numbers will perform better.
Note that when you install you will need to remove Nvidia drivers/software and install AMD software. Not doing this correctly can cause problems.