There may be many things causing the car to stall upon acceleration. You say that the car starts right up after stalling and this seems like an electronic problem. When you accelerate you step on the air and the air-flow meter drops, tells the computer to reachen the fuel. If your potentiometer (air-flow meter's variable resistance meter) is bad, it will have a nonlinear resistancecurve. The bad potentiometer will make the injection control computer mad and it will shut down also shutting down the car. This happened to me and right now I have the potentiometer disconnected. Every time I connect it my car stalls upon acceleration. You will have slower acceleration (no burning rubber) but no other problems with it disconnected.
Fuel pump relay can also be causing this problem. Try jumping pins 87 and 30 of the fuel pump relay. This will make the fuel pump work all the time. Then, drive the car and see if the car will stall.
Here is how the parts look.
Red circle - potentiometer,
green circle-fuel pressure regulator.
Hope this helps.
1989 230E, 8v, 166.000 km, updated to 94/95 trunk & hood
2002 Daewoo Nexia 50.000 km
1987 VW Jetta GLE 16V, Recaro seats
1982 Volvo 240 DL (lovely car!)
and few more american cars.