When temp. hits normal RPMs go to zero and car stalls/stops running.

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I don't know about car mechanics. I don't know about diagnostic codes, etc. My car battery was worn out but at the time I could not afford to buy a new one . I work from home and I often do not use my car for a week at a time. When I needed the car after it had sat for several days, the battery would be dead and I would get a jump from someone in another car or sometimes I used the 12V battery of my tire inflater/emergency light compressor to give my battery a jump/boost with my jumper cables. The car would start right up. The car would run perfectly and if I had to stop someplace and turn the car off for a while (to do the groceries, for example, for one hour), it would usually start back up without needing a jump. I had to jump it on about 10 separate occasions, total, before I got a new battery. After about the seventh time of jumping the car, the car would start with he jump first time that day, as usual, but then, while the engine was running/driving or idling at a light or in stop-and-go-traffic, the RPMs would drop to zero and the car would stall/stop running. It would sometimes start up again without needing a jump, but stop again shortly thereafter and then need a jump to start. This happened on the day that I was on my way to buy the new battery, and it happened again when I left the shop with my new battery. I went back to Sears, where I bought the battery, and the tech said that he had seen that happen with other cars of various makes; that after putting in a new battery the cars would exhibit the dropping off of the RPMs to zero and stalling while driving home from the shop. I explained that my car had done that just before I came to Sears that day. The tech had no solution for me. Ever since the time mentioned here, the car has continued to stall while in in use, and I have given up using it except for short drives limited to a few miles from home. I have noticed a very significant symptomatic aspect. It has to do with the vehicles temperature. When I start the car first time that day, and the car is cold, meaning the temperature at rest, even in Miami 70-85 degree weather, it starts up fine and runs perfectly, strongly and smoothly, with no hesitation or backfiring when I accelerate sharply/floor it, THAT IS UNTIL the car's operating temperature reaches the normal mark on the temperature gauge, at which time, the symptom of the RPMs dropping off to zero and the engine stalling occurs. This pattern happens without exception. It is always the same. I now have stopped using the car and I only take it for rides around my neighborhood to keep it from being damaged by non-use. I always circle back near my home, to assure that when the temperature is nearing the normal mark I am practically in front of my apartment, because once the car reaches normal temperature it stalls and it will only start once or twice more and then no more until it is allowed to cool down. After the first or second stall it will only take me a block or two before it stalls for good, until it cools off, hence, why I need to stick close to home. If I allow the car to cool down to resting temperature, then it will start again. Then if I let it run, as soon as the temperature gauge is pointing to the N (normal temp.) then the stalling occurs again. And this pattern has not changed at all. It is always the same. A friend Jaguar mechanic came to my home to check out my car and he felt that it must be the fuel pump, then he consulted with a Jaguar mechanic friend of his and the friend said that it sounded ike the ECU. Later, my mechanic friend wondered if it might be the fuel pump relay. Please help.