If you have a home battery changer, You may also want to try an over-night battery charge [ low setting]..
That should get a possible low battery voltage back up for further testing. When battery is fully charged , give it another go......
This would be better than a jump and you won't loose any memory.

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A Dalton

I agree with Steve and others. On my E230 wagon, I had your exact problem and a new Mercedes battery fixed it.

I was surprised how many amps the system must pull to turn over.

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2004 E320 4matic
2000 E320 wagon
-- in the past---
1992 190E 2.3 5-speed! (European Delivery Baby)
2000 ML320
1978 240D - 340k mi

Same problem on our 2003 e320. While at the light, the car stalled & turned off, doesn't want to start. All electrical components work (light, radio, alarm, etc). Speculating it's the fuel pump. Not sure yet, since it just happened a few days ago & the new pump is on it's way.