It could be the mic’s wire or the mic, but even a standalone mic wasn’t working. It worked OK with Audacity, which uses Alsa.
It turned out the problem was that the Sound control panel in Gnome didn’t work. Somehow, the system decided to change the input from Analog to Digital SPDIF. I don’t use SPDIF.
The fix was to install pavucontrol, which is a more elaborate volume control than Gnome’s. Gnome’s only detected the digital audio input, while pavucontrol detected every type.
So, the first thing was to switch the configuration of the sound card to use ANALOG for both input and output.
Then, I ran Skype. It didn’t work. It turned out the fix was to go into the Applications tab on pavucontrol and then change Skype’s input. To do this, you need to be making a call, so Skype opens up the mic.
This is so complicated, I really don’t know what to say. I suggest getting a USB headset so it just shows up as a pair of devices in the audio menu.