Well, From the sound of it, I will assume your network cable is a crossover and goes from one PC to the second, with no hub or switch in between. In this case, the only way to get rid of the icon is to "hide" it using the start menu's "Hide Unused Icons" feature. You can go into your start menu properties and choose to "Always Hide" the network status icon. The only other way is to make sure that you do not have "Show Icon When Connected" checked in your connection properties. And even if you do, if I remember correctly, you will get that icon to let you know you're not connected.
EP, glad to see you come back and tidy up...did want to ask a one day favor, I want to enhance my resume , was hoping you could make me administrator for a day, if so, take me right off since I won't be here to do anything, and don't know the slightest about the board, but it would be nice putting "served administrator osnn", if can do, THANKS