Had a problem like that on a different make of car and it turned out to be a vacuum tube on the automatic idle control. Not a big cost...just need a good mechanic. I am not saying that this is your problem, but it could be something with the fuel injection system.
I kind of have the same problem. Except mine is more that it wont start untill after you let go of the key and try starting it a second time, but i have also have had the problem that you described. GM has told me that it is the fuel pressure regulator, and because the midwest has ethanol blended fuels it gets eaten away and leaks. I dont know if your from the midwest or if you have ethanol in your fuel, but try replacing the fuel presure regulator. I wouldn't go with a GM part because the ones they have put in my car dont seem to last that long maybe 8,000 miles.
My wifes car has just started doing this (2000 SSEi) but hers doesn't shut off. When it'* started the car shakes for a minute or two then clears up (usually after putting in gear). I getting ready to take it in to get her rear window and leaking trunk fixed so I guess I'll have to get this looked at as well.