My engine shut off while driving can you help me fix it?

It checked out fine.

And this is what I found. I fixed it and the car runs just fine. Now it was melted by the EGR valve.

I have the car listed above that stalls randomly it has been doing this for at least a year. I was told by a mechanic that he thinks the problem is a body control module. He checked the wiring and said it was fine. He said that he has done some research online and found that a lot of 2001-2005 Impalas have had problems with the BCM and that replacing it has fixed the problem. I have no reason to think he's lying about it, but I don't want to spend money on a part I don't need because I have already replaced the cam and crank sensors and the car still stalls. Any ideas would be appreciated.

Recently I purchased a used car so I haven't own this vehicle long. The car has run good up till this point. I started the car normally drove to the gas station. I shut the car off, fuel, started and drove 32 miles for for stopping. The car was idling fine when I came to a stop. I got out checked my trans. Fluid which was good listened to the engine and shut it off. Everything was going good until I went to start my car again. After about 4.5 hrs I went to start the car. It started and then stopped right away. The only way it would stay running is by giving it a little press on the accelerator when starting and keeping the RPM's up. I did have my cell phone charging while the car was off the whole time however the car did turn over fine. I drove the car 15 miles and all the way ever time I'd slow down it would want to die and I'd have to start it and give it a little gas. As long as I kept the RPM's up it ran good. What could this be?

There is no check engine light on. The car runs fine as long as I control the idle. It doesn't take much to keep it running, it's more annoying because it will dye if I don't put my foot on the pedal to give it a little gas. It doesn't surge, hesitate, stall at highway speed. Only when the RPM goes to idle it doesn't level out.

Could the idle air valve motor be bad?

Here's something to throw you off. I did take apart the throttle body, well just the boot and sprayed throttle body cleaner in it. It was very dirty. I took out the IAC motor and clean it and in the port where it goes. It looked like the valve was all the way out, but I don't know if this is good or bad. I put it all back together, same result, won't idle. I disconnected the IAC motor as you suggested, no change. It was late so I quit fooling with it. Woke up this morning and same thing, won't idle. Made it the 45 miles to work o.K. And parked it for the 9 hrs at work. When I went to leave I started while I stuck my head out the window. I heard a small, wheeee sound like a little motor and it is now idling just fine. The wheee sound was like a little fuel pump in the engine compartment and didn't last long. I did hear the fuel pump too so I know it wasn't that. I did some other things tonight on it, flushed coolant (need it anyway), checked fluids, drove it. It is working good but makes me wonder what is wrong. I will get the codes checked and probably replace the IAC motor. I have a friend that his brother works at a chevy dealer so he can get me a part at his cost. Any other ideas!

Thanks, you were a lot of help. I figured that is how the IAC worked. I didn't know about the sensor on the tip though. I think what was happening is that the motor was stuck closed and it wasn't letting any air in, that's why the car was dying and I can to step on the accelerator to keep it running. Now that the motor is work it's letting air in. I don't know if I should replace the motor anyway or consider it fixed. I found the part online for $50 but I do have a buddy who can get it for me from a dealer cheaper. I'll see how this goes, I just don't want to work on the car this winter in the cold if this happens! My luck it would be -40 °F that day here in Minnesota! Thanks for your help.
What is your thoughts about replacing the motor anyway, should I buy one and get it done or is it fixed?