Hello everybody,
Thank you so much for all your replies! It is so good to know there
are people out there that know how to help!
My wife asked me to reply myself about these technical things.

We bought this "Jukebox Recorder 20GB" in 2004 from the States. It
came with a built-in Li-Ion battery pack, so there is no possibility
to change the batteries at all (as in 4 AA-batteries). I always
wondered how this would work once the batterie pack dies. As I
remember, Archos promised the battery pack will last the whole
Jukebox's life. It is built-in, so you can not even change the pack!
The power supply comes also from the States and does have a heavy
transformer in it. In Germany, we used it together with another
transformer which turns 220 to 110V. However, the power supply is
not the problem, since we almost never used it at all, but used to
charge the batterie pack via USB on the computer.
Right now, when I connect either the power supply or via USB with
PC, I can hear a very low sound, it is like buzzing. That means
there must be some power going into the box. But nothing happens to
the Jukebox at all: No lights go on, no backlights, it appears
absolutely the same as if it was turned off, except the buzz.

Now, what I think might be the problem: I imagine that the outside
power goes "through" the batterie pack, and after this into the
device. If the batterie pack died, the power does not even reach the
device itself. Could this be the problem at all?
Can I just find a new batterie pack and exchange it? Any experience
with this?

Thanks again,
Ralf