I find that the easiest way to do that in Illustrator is by using Filled shapes. There’s a tutorial here that shows the concept:
In your case you will make one shape for the frame, then finish joining your text in that one spot where it overlaps to make it one shape. Then join the text to the frame. (It’s just several Unite steps, once you have the filled shapes created.)
Your frame will have the center rectangle cut out from the rear rectangle, with a fill in between. (Use Minus Front with the rectangles given a fill…it creates a compound path.)
After the frame is created, you can use the Unite tool in the Pathfinder palette to join everything together.