Q&A

Although I am a "Rookie", I give you some possibilities:
1. If you are not using the original Saturn key, that may be your problem.
2. If you are using the original Saturn key, be sure that the fob of the key is not being "sheilded" from the steering column by another key or something on the key ring.
3. The fob of the original key has a microchip in it that sends a signal to a receiver in the steering column. If something disrupts that signal, the engine will shutdown after about 10-15 minutes. If your problem is not either 1 or 2 above, it could be a failure in this system. Have a mechanic scan for codes with an ODB II scanner ... or, do as I did, buy an ODB II/EODB code scanner and read the codes yourself. For $100-$200 you can buy an ODB2/EODB scanner that reads generic (SAE) codes like: P0,B0,C0,U0, etc. but manufacturer specific codes like: P1,B1,C1,U1, etc.
Good Luck!

my saturn did the same thing, I just had the CPS (crankshaft position sensor) replaced at it runs perfect now