I had the same problem. When the headlight switch is turned to the position to turn the dome light on, the ground contacts for the dome light are supposed to touch at the switch to complete the circuit. My contacts weren't touching and simply needed slight bending to correct. As stated above, you can check if this is your problem by completing the ground with a jumper wire at the switch to see if the light turns on. The repair is easy, but does require removal of the switch to get at the contacts. Easy does it when doing the bending.
