1998 Chev Malibu starts then stalls

The most likely cause for the problem you are describing is the anti-theft system. GM calls it a passlock system. Your car has a 'key-reader' mounted on the ignition lock cylinder (inside the steering columnm, or inside the dash, depending on where your key goes). This 'key-reader' sends out a signal to the body control module indicating that the proper key is in the ignition. Note: this style antitheft system does not have 'chip' in the key. These key-readers fail frequently and send a an incorrect singal the the module, where the car will start and run for a few seconds, then stall. An easy verification of this problem is to watch for your security light to be flashing somewhere on your dash. The passlock cylinder is only available from the dealer, and while it is a fairly easy part for someone like me to change, I do not recommend it for someone who is not a mechanic. Additionally, even if you do change the passlock cylinder, you have to perform a re-learn procedure or the car will never start. This probably isn't the answer you wanted to hear, but better that you know. Hope this helps!