It's not possible to buy change the PCI slot to a PCIe slot without additional hardware; because PCI and PCIe are different standards everything about them is different: the way they transmit data (serial v. parallel), their encoding scheme (PCIe uses 8B10B), different hardware controllers on the motherboard and PCI card, different power supplied to the card.
http://www.newegg.com/Product/Product.aspx?Item=N82E16815158190
This is a card which converts a standard PCI slot to a PCI Express x1 slot. Functionally, this will do what you are asking.
What's the catch then? I do not recommend you do this. These converter cards are for edge cases. Based on the motherboard you a citing (and the fact that it does not have PCIe), you could get a suitable replacement for a modest sum (less than $100-200), which will have PCIe. The problem is that no one can guarantee this solution will work. First, most graphics cards use the x16 slot; this is an x1 - meaning you are figuratively cramming something designed to fit in a big hole down a much smaller hole. Second, the power draw issue - the PCIe spec is designed to provide several times more electricity to the card than the PCI spec was. Because the adapter runs through a PCI slot, you will be capped at that bandwidth.
Third, and most importantly, performance. I get the impression that you are upgrading your GPU because you want better performance. You've got to understand that when using an adapter like this your performance will be *terrible*. It will most likely be *worse* than what you have now.
Fourth, it is possible to get a newer card for your current motherboard standard (AGP), but this is a standard that is mostly unsupported. I would recommend looking into this if you are budget-concious, as this may be the most affordable route. The only concern here is that you may not got any real performance increase over the card you currently have.
Your best bet is to do an upgrade of the CPU+GPU+Memory+motherboard. Probably not the answer you wanted to hear - but it's really the only viable option.