This is another shameless re-posting of a solution published by others. See e.g.
Why am I doing this? Mostly because this blog doubles as a notebook where I can write down solutions that worked for me so that I don't find myself scouring the internet over and over again for the same solution.
I hope that I'm adding a bit of value by providing step-by-step solutions and snazzy screenshots though.
even though the camera is working fine there's only a completely black picture in skype (not underexposed -- just plain black):
Use LD_PRELOAD to load the 32 bit V4L library (which you need to have installed -- use apt-file to find out what package). You can either start skype from the command line, or make changes to the launcher, either via main menu as shown in the screenshot or by editing ~/.local/share/applications/skype.desktop i.e. run skype with
env LD_PRELOAD="/usr/lib/i386-linux-gnu/libv4l/v4l1compat.so" skype
At this point starting skype + webcam should be a happy marriage: