iMac showing "Built-In Ethernet" as inactive

Use System Preferences > Network Panel and make the Ethernet active again. If you can't then there may be a hardware fault.

Sadly I don't have a 10.4 machine so I can't remember exactly how it is done. I think they changed this panel in Leopard.

Edit: Had a quick search
1. Choose System Preferences from the Apple menu.
2. Click Network.
3. Choose “Network Port Configurations” from the “Show:” pop-up menu.


4. Make sure the checkbox next to Ethernet is checked.


5. If it is not, check it and click Apply Now.

Edit 2: For those with Mac OS X 10.5 (Leopard), it's easy. Just choose: Make Service Active from the tool menu and click Apply. The old network controls were a bit of a mess.

 

I've got an additional question along this theme-

If the built-in ethernet doesn't even show up as an option anymore in Network status/preferences, is that a hardware problem most likely?

 

Not necessarily the case. That list can be edited, so you might be able to add ethernet back. If you still can't then I guess it's a hardware fault?

Are you on Leo or Tiger? For Leopard I can give instructions - for Tiger I'd need to Google.

 

did you try restarting first?

 

Are you not seeing it in step 3 above or step 4?

Because if it has been unchecked in the screen corresponding with step 4, it will not show up in the overview shown in step 3. If it's not in the checked list in step 4, then there's more of a chance that something is physically wrong.

If the previous two suggestions don't help, I'd try creating a new location (by clicking the "new" button in step 4 above) and see if it appears in the list there, just to rule out a screwed up location.