Thanks for the link and your advice.
I screwed duet onto the base of a printed case made from PETG. I did that on my computer desk, no clutter, no parts – clean surface in a buero, not a shop. Chances are very low that something conductive made its way onto the board.
I gave it a blow front and back with an camera blower. I removed it from the case, inspected front and back visually. And I also swapped the fuses for new ones. (The ones it came with look fine, but I just wanted to be sure. I don't have a multimeter at the office.)
The result stays the same. When I plug the Mini-USB in, the LED next to the connector glims so feint that it is barely visible in daylight. The Mac reports that too much power is drawn and the port isn't mounted.
I you could advise me on the points and expected readings, I should be able to measure the board. But I don't have the expertise to do that with the info I find around this board.
So, if there are no good ideas otherwise, I'll try the warranty.