out of frustration, I havw just put the old, not so good working battery back, phone works fine. So something is wrong with the battery. Yes, its the correct one for 3GS, even the APN numbers are identical!!

I replaced the battery, and when I switched it on again, the battery level showed 100% and than later goes in start up mode again or asks for activation by connecting to i tunes. As soon as I take the connector from the charger out, the phone dies. I also tried to connect the phone directly to the computer, but there it does not start at all!! Any suggestions?