Christy! You and I have been having the same problem. I have been using Premiere for over a year now and have never figured out how to handle the black bars, but I think I just figured it out.
I tried a lot of things and I don’t know exactly what handled it but for the first time in a year there is no black border around my export and the source is the same as the output! Here’s what I did:
Make sure you know exactly what size and format the video was shot in. It took some tinkering but I figured mine as:
640X480
29.97 fps
Square Pixels (1.0) 4:3 ratio
No Fields (Progressive)
Create a new project.
Go to Custom Settings and make sure all the settings match exactly before hitting OK.
Once the project is created, go to Edit>Preferences>General and turn off Default Scale to Frame Size.
Import your clip, and it should fit right in the window.
Let me know how it goes!