Car dies when slowing and stopping

I am driving a 91 Chevy Beretta GT. It is my first car. It has started to sputter and die whenever I brake and stop. It also seems to do it when the engine revolutions drop below 1000. I replaced the fuel filter and it worked fine for 1 day before the problem returned. Could it be clogged fuel injectors? Or the fuel pump?

Idle speed is controlled by the Idle Air Control (IAC) valve. If this valve is dirty, worn out, or otherwise compromised, the engine will not idle correctly. Sometimes cleaning the IAC valve works wonders.

With the engine warm, and idling in P or N, press the brake pedal. Let off. Does the engine behave differently when the brake pedal is depressed? If it does, the brake power booster could be ruptured and losing vacuum.
You need to check the fuel pressure. A dirty fuel filter (which you changed) can cause fuel pressure to drop. Other things can cause fuel pressure to drop, also. Results?
You want to clean crud from the idle air control valve pintle and the intake manifold passages that lead to it, and from it. http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c152800a73e3

I can’t tell from the description, but I am thinking it may be the transmission locking up as well. The description is just not clear enough for me. I would say it is likely the transmission or the IAC.

The problem seems to be happening intermittently for me.

Depressing the brake pedal does not seem to affect the engine behavior or sound

You could have one of the more normal problems from the old says, although your car wasn’t prone to this problem: Worn out wires on the ignition pickup if you have a distributor with vacuum advance and a manual transmission. The manual transmission means that the vacuum advance moves the pickup plate every time you shift. You may not have the vacuum advance. You could also have a slow moving EGR valve due to carbon buildup. Neither problem is very likely, but may be worth looking at. The idle speed control motor could be too old to work right.

sounds like the torque convertor lockup switch which disengages or is supposed to the torque convertor and allow slippage at idle, without disengagement the engine will die as it slow and stays locked up.

My Beretta died on me when I was driving on the interstate today (4/22/09). The engine just stopped running. There was no sputtering or any other extra noises. It would not start when I tried to start it after it died. It tried to start but would not turn over. The two guys who have been helping me with the car think it is the fuel pump and I am going to check that. Is there anything else it could be?

The automatic transmission seems to be shifting fine.

When fuel pressure falls off, the engine usually sputters to a stop. It would be nice if you could use a fuel pressure gauge. If there is a valve, which looks like a tire valve, on top of the engine, that is the fuel pressure test port. At least, remove the valve cap and depress the center of the valve. If there is fuel pressure, fuel will squirt out.
Use a spray can of Starting Fluid in the intake tube. Spray a two second burst and attempt to start. Results?
Swap the fuel pump relay for another of the same kind. Results?
On GM vehicles, there is a loose wire for testing the fuel pump. It’s a single wire, just “hanging around” near the fuel pump relay (which may be in a plastic cylinder near the battery). Use a hot jumper wire to this “fuel pump prime connection”. Results?

the car tries to start when the starting fluid is sprayed into the the intake but it doesn’t keep running. The fuel does squirt out of the fuel pressure test valve. The fuel pump does run when I connect the battery to the hot jumper wire.

It was the fuel pump relay and some worn out vacuum hoses.

The cause of the sputtering was the “fuel pump relay and some [bad] vacuum hoses”.
Did you test the fuel pressure and find it was erratic, or use some other method to determine that the relay was intermittent? Intermittent problems can be VERY difficult to chase down. This is one reason there can be no STANDARD solution to a particular problem.