I have a 98 Civic LX with the D16Y7 engine. A few days ago, while driving at night, I noticed the dash lights were getting dimmer and dimmer. I stopped to investigate but while I was poking around the engine died and there was no battery power to restart. The AAA guy jump started the car but it would immediately die when the cables were disconnected.
After towing it home I pulled the battery and the alternator (PITA) and took them down to Autozone for a diagnostic. The Battery tested bad and the alternator tested good, so I bought a new battery and re-installed the original alternator but after starting the car there was no output from the alternator.
I the took the alternator out AGAIN and took it down for a re-test. It tested good but I decided to buy a new one anyway but when I installed it there was still no output. I checked fuse 15 (good) and the wire harness (good). I noticed that the check engine light was on so I pulled the codes and got P0135 and P1298. I don't know if this is real or its just caused by the electrical problem.
Any Ideas?
Here's a new wrinkle. I just checked the alternator connector and there is no voltage on the black/white wire. I think there's supposed to be 12volts with the ignition on.

 

This really sounds like a fuse 15 problem. Check the wires behind the engine where they go over the intake manifold bracket.

Does the "battery" light come on when you turn the key on but don't start, and go out after starting?

 

No, The alternator light doesn't come on at all

 

That means you're not getting power to the alternator on the "I" wire, like your test showed. Need to check the wiring, also make sure you were checking the right fuse.

 

Apparently my fuse counting skills aren't good. When I pulled the actual fuse15, I found it to be blown. After replacing it the alternator started working (14.5VDC). But now the Battery light is on dimly all the time when the engine is running.

 

It's prone to blow again unless you get under the car and tape up the wires. This is a well-known bug with the 1996 to 2000 models.

Light coming on dimly probably means the alternator is somewhat bad. Or the wire for the light is touching ground.