A high-pressure discharge lamp and a reflector lamp including a discharge vessel enclosing a discharge space which is provided with an ionizable filling comprising one or more halides. The discharge vessel is substantially constituted by a ceramic material having first and second end portions. Current-supply conductors issue through each end portion to respective electrodes arranged in the discharge space so as to maintain a discharge. At least one of the current-supply conductors is formed as a rod including iridium. The rod is directly sealed to the ceramic material.