This is normally done by configuring Network Address Translation (NAT). If your router has a NAT option, you set up a local network (ie, 192.168.0.0/24) where all your computers reside and those get translated to a single outgoing IP. The down side of this is that your individual machines can't be reached from the outside world directly, even though they can connect out to anywhere they want.
I any one of your machines are running some version of Unix (BSD, Linux, etc) and has two network cards or could have a second added, you're in luck. If they're all windows, there is some internet connection sharing stuff that might do what you want, but I've never used it and couldn't tell you anything about it. Either way, the machine that's doing NAT for the other machines would have to be on all the time when you want the other machines to use the net connection. If you have any questions on the Unix side, let me know.