For several months I had had problems with downloading data from my PC. I thought it was due to some monitor settings (I had bought a new monitor about then), and that I had to live with retrying several times. But later it got worse. I could not download anything.

Then my display got dim, and then it became blank. Of course, that happened at a most inconvenient time, when I was in Italy. :-(

Here's what happened:

Replacing the battery itself was no big deal, thanks to the instructions published on the DataLink Developers list. I took the whole battery-holder out first, together with the red cello tape. Then I took the battery out from the holder, which enabled me to not remove the tape at all, I just had to loosen its grip on the battery itself. Then I put the new battery in, snapped the holder (with the tape) back in, and used the reset contact, as instructed by the PC software help file, and the red tape message.

That was something new to me, did anybody mention it before?
Also, there was a thin white plastic ring, as shown below. I guess its purpose is to separate the back and the watch's inside, ensuring there's space for the sound to be generated in.

The only problem was the back. I could not put it back in place.
As far as sound goes, I am pretty sure the only thing to watch for is the whitish round piece of metal glued to the back (see below), which must be in contact with the spring in the watch itself.
(On the larger picture, you can see the back was made in China. ;-)

So, I held the parts together with a rubber band. At that point, I have downloaded all the stuff back into the watch, with no problem.

I took it to my office, where the back was put in place using the socket/C-clamp method.

Thanks for your help, everybody!