It sounds almost like a vacuum leak but I would assume a code would pop up for that.

 

Had the exact same problem, with ac off and car warmed up coming to a stop the car would drop from about 850 rpm to 500, when you turn the ac on the idle returns to normal.....after pulling out much hair, I found the fuel filter\regulator was the problem, it is serviceable on the contrary to what every dodge dealership has told me, if you look up the part as a fuel pressure regulator on the advance auto parts web site you will find the part(***part number at advance 24026****), its 50 dollars, relieve fuel pressure before doing anything, pull the fuel pump relay and start the car and wait for the car to stall out, now on the fuel pump is where this baby is located, disconnect the electrical connectors and the fuel line, be careful not to damage the plastic clip on the fuel filter as you will need to reuse it on the new filter. there is a tab\metal tang on the side of the filter, press the tab and twist the filter and pull out, carefully remove plastic locking tab and replace on new filter, lube the o ring with a light oil, push the new filter in and twist it until it locks in place***filter will only go in one way so pay attention***, reconnect electrical connector and fuel line, reinstall fuel pump relay and start car.....peace out weird idle

 

call the Dodge dealer and ask if your car needs a flash update for rough idle issues. You will need the VIN for him to check it out. There is an update for the 2000 models and iirc it includes the 2001 as well. There is also one for harsh a/c engagement. Won't take a few minutes to check and do the reflash to the ecu......results are priceless!

 

I spent a lot of wasted time and money at my Dodge dealership and other local mechanics trying to fix this problem. Like others have said before me my 2001 Neon would shake at stops when the idle would go to low. Putting it in neutral or park would sometimes help. Other than that the car ran great. No codes would ever come up, so the car would always pass diagnostics. The dealership would flash the PCM when they didn't find anything wrong or play parts roulette. If you have already had to spend money on parts and service without having your problem fixed you may want to look at this guy's webpage

http://www.cardomain.com/ride/464248/2001-dodge-neon

He describes the exact problem that I was having and shows what the dealership did. It involves you drilling an 1/8'' hole in the throttle body as shown in his pictures. This wasn't much of a risk for me since I still had the old throttle body that the dealership had replaced with a new one. This fix worked and since the throttle body position sensor is already at home position when you're at stoplights the computer can't adjust to the extra airflow. When you've already got your foot on the gas pedal that little hole doesn't seem to make much of a difference when the throttle starts to open up. Obviously this isn't the best fix, but let's face it if you haven't been able to fix it by the time you read this post this might be what you need.