A device for lighting a discharge lamp has a drive circuit to feed alternating power the discharge lamp, and a control circuit. The control circuit controls the drive circuit by a drive pulse to perform a burst dimming control over the discharge lamp. The control circuit has detector, subtractor, a digital filter, and pulse generator. The subtractor subtracts the detected lamp current by the detector from a reference value. The digital filter integrates the output of the subtractor as an integrator. The pulse generating means generates the drive pulse based on the output of the digital filter. The lighting time period has a first time period immediately after a start of the lighting time period and a second time period following the first time period. The control circuit sets the reference value to a target current value in the second time period. The control circuit increase the reference value in the first time period to the target current value until an end of the first time period. The digital filter retains the output obtained at an end of the lighting time period until a next lighting time period starts. The control circuit adjusts the lamp current to the target current value during the lighting time period.