A high-pressure discharge lamp assembly has a discharge lamp (1) and a concave reflector (11) arranged around a longitudinal axis (30). The discharge lamp is closed in a gastight manner and comprises a first and second end portion (3, 4) and an ionizable gas filling, and in which a pair of electrodes (5, 6) is arranged. A first and a second current-supply conductor (7, 8) are connected to the pair of electrodes and issue to the exterior of the discharge lamp (1). The first end portion of the discharge lamp extends through an opening (14) arranged in a center section of the reflector. A conduction member (9) is connected to the second current-supply conductor and extends through the opening in the center section of the reflector. The conduction member is connected to a contact member provided on a surface of the reflector facing away from the discharge lamp.