The engine idles beautifully. However, the engine hesitates upon acceleration. Once the engine reaches approximately 2000 rpms the engine smooths out and runs well. This is even more noticeable when the AC is running.
Carburetor or injection? Year?
When did you last change the plugs, wires and dist cap?
If simple maintenance such as replacing the plugs, plug wires, and distributor cap does not help, or if they are relatively new, the next item to look at would be the Mass Airflow Sensor (MAF). The MAF may need cleaning or replacement.
And if the things VDC mentions don’t help the next step is a compression test.
Does this engine have adjustable valve lash?
A vacuum leak could be a suspect or an air leak in the intake tract between the throttle plate and MAF sensor could also cause this; all assuming the MAF is working properly.
This would not be one of those modified Neons would it? (as in aftermarket air cleaner and so on)
Yes, all the above. And an EGR valve that does not seat fully can be a vacuum leak that is very difficult to find. And a Throttle Position Sensor will sometimes fail in the off idle range.