Hi Hunter,
Are you comfortable opening up your computer and touching the electronics, possibly disconnecting cables, and pulling circuit boards out of one slot to be replaced by a different circuit board? If so, sure, you can handle it. As an alternative, it would probably be a small bit more expensive, but easier to buy a USB modem. Try something like this:
http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=3718719&CatId=564
You could just plug this into an available USB port, unplug the phone cable from your existing modem and plug it into this. That should do the trick.
As a third alternative, since you hate dial-up, have you looked at broadband? I'm assuming you don't have cable or DSL available in your area or you would have already switched. What about cellular internet? AT&T, Verizon, Sprint, and others all offer small USB "dongles" that connect to their respective cell networks and provide a much faster speed than dial-up. Sometimes faster than DSL. I have used both Verizon and ATT. I prefer VZ in my area, but you should check your coverage. I think they all cap the monthly download size at 5GB, but that should be more than enough. If you're coming from dial-up, you're not downloading 5GB per month, anyway. Just don't go crazy with online video and downloading huge podcasts or something.
Good luck!