There is a double-size box and on one side a modern dimmer switch. The other half of the box is empty and I want to put a new double three-pronged electric outlet there. There is one lead-in into the box with one white, one black, and one bare wire coming into it.

sounds like you have no electricity live in the box. What you have is an interupter wire for the light.Technicly it should be 2 black wires. What it is , is the power wire Black from the light is run through the switch. In your case the electrition just used a standard wire and used the white wire to return power to the light when the switch is on. the bare wire is the ground wire and should be connected to the box, some dimmer switches have no place to connect a ground. In any event you dont have a white common wire, witch you need. the white wire on the switch should be black and a good electrition will use a black sharpie to color it for safty reasons. P.S the info you got before mine will be a good way to blow the circuit, wrek you swich and light and possible fire
one black and one white in box is a switch loop power is at light box. To put a plug at that location you need power and neutral at that switch box. you need to have an electrician or someone capable get power to the two-gang then they can then use switch loop as a switch leg for fan as well as give you power and a neutral for the plug.. If you have to ask don’t do it. safety first!