A high-pressure discharge lamp has an outer envelope (1) in which a discharge vessel (11) is arranged. The discharge vessel encloses a discharge space (13) with an ionizable filling. The discharge vessel has two mutually opposed neck-shaped portions (2,3) through which current supply conductors (4,5) extend to a pair of electrodes (6,7) in the discharge space. A lamp base (8) of electrically insulating material supports the discharge vessel. The lamp base also supports the outer envelope (1). The outer envelope encloses the current supply conductors and is connected to the lamp base in a gas-tight manner. By controlling the atmosphere in the outer envelope, a simplified and compact high-pressure discharge lamp is provided with an accurate positioning of the discharge vessel with respect to the optical axis of the lighting system. The high-pressure discharge lamp can be suitably applied in an assembly with a reflector.