Charging cable does not charge the battery anymore

Alain Rouquet your measurements of amperage and voltage is from the power supply transfomer or the door bell itself? where did you measure?

The measurement are made on the cable supplying the Ring.

this cable has a total resitance of 2,1 ohms (telephone wire)

On the transformer terminals, there is 9,65V under this load,

then I have 2 resistors of 10 ohms each, which drop 3,92V each, so I have 1,8V at input of the cable, and by calculation about 1 V on the Ring terminals.

Hi Alain Rouquet.

What is the transformer voltage with no load?

Have you measured the voltage at the Ring terminals?

According to the info on the battery for your doorbell (assuming it is the same one as featured in the ifixit link) the battery requires 8-24VAC or 5VDC @1A and according to your measurements you are supplying about 3.9A.

Post some close up pictures of the motherboard so that it can be seen if there are any obvious problems.

Here's how to do this Adding images to an existing question

You probably don't need this but here's a link to the ifixit Ring Video Doorbell Motherboard Replacement guide which may be of some help in gaining access to the motheboard

0,39 Amps, not 3.9 Amps .

I will disassemble the Ring in October to check what went wrong on the mother board, and either repair this AC supply or arrange a 5V DC supply through the USB charging...!