aethaniel,
thank you for your reply.
the 2V battery was the initial object to be measured at the analog pin, the power was supplied by USB-5V!
After first indications of incorrect results in measurement, I simplified the circuit to only test with a resistor-divider (as described below) for analog measurement and USB-5V supply, over which also the result of the measurements is shown via the serial monitor at the PC (serial.print).
The unexpected thing is, that the readings drop as described in my previous postings and after a Due-reset or restart of the serial monitor (which also restarts the Due) the first readings are correct again, before the measurements drop again to sigificantly smaller wrong values. For me, this is an indication that there is something wrong with the analogRead command, not a hardware problem.
It would be helpfull if someone could confirm. The only hardware needed for a test is a resistor of approx 2k-ohm connected from 3,3V to A0 and a second resistor of half size from GND to A0. This circuit will apply approx 1,1V to A0 (1/3 of 3,3V). Running my test-program should demonstrate the problem at the serial monitor.