An electrodeless discharge lamp device comprises an induction coil, a pair of cores 1a, 1b, and thermal conductor 2. The cores 1a, 1b are configured to generate a high-frequency electromagnetic field for exciting a discharge gas with receiving a high frequency electric power. Each of the cores is arranged to give an overall circumference around which the induction coil is wound. The thermal conductor is formed into an approximately cylindrical shape and inserted into the space so as to be thermally coupled with the cores for radiating heat generated at the cores. The core has its inside face at least center portion of which is spaced radially from the thermal conductor by a predetermined distance.