1997 Chevy Cavalier ingnition start problems

Electrical problem
1997 Chevy Cavalier 4 cyl Two Wheel Drive Automatic

when engaging the key ignition to start vehicle, if you let off the ignition the vehicle stalls. If you hold the ignition on, just before engaging the starter, the vehicle will stay running. We've tried chaning ignition cylinders, starting the vehicle with the key ignition out, have double checked all wiring and fuses, and are at a dead end road. Please help. Any tips or suggestions would be appreciated. Thank you