I have an 89 Honda Accord Lxi. Tonight I drove about 2 miles to a local store, no problems. When i come out of the store the car is completely dead. Not even getting the clicking when you try and start it. I immeadiately got concerned that it wasn't just the battery. So i have someone come and jump the car. It starts. As i'm driving home i notice the dash lights are all dim. When i got home i sat outside and let it run hoping to charge the battery. While i was sitting there I noticed the lights would get brighter when i revved the engine. After a few minutes the car just died and all the lights went off, needless to say i could not start it after this. All of these things lead me to believe it is not the battery. Anyone have any words of advice. I'm thinking possibly the alternator?

Welcome to AF!

Definately sounds like the alternator. This is a stupid question, but is the belt still there? I ask that because some people might not think to check.

Starter gone? Maybe another possibility

this could very well be the alternator
check pulleys\belts

My dad had a similar problem, except the car just shut off completely while we were driving. I believe it was either the alternator or the starter.