The production of O-2-enriched air and of high-purity oxygen was studied in a flow-through and in a dead-end permeator with BaCoxFeyZr1-x-yO3-delta (BCFZ) hollow fiber membranes at high temperatures using (i) pressurized air/O-2-enriched air as feed, (ii) reduced pressure on the permeate side, and (iii) a combination of both. For the effective production of oxygen with a high space-time yield, a two-step process is proposed. In a first step, O-2-enriched air with an oxygen concentration of 30-50 vol % is produced in a flow-through permeator from I bar of air as feed applying a vacuum pump to keep the pressure on the O-2-enriched air side at about 0.12 bar. In a second step, pure oxygen is produced in a dead-end permeator using the O-2-enriched air from step 1 as pressurized feed and using again a vacuum pump to establish a reduced oxygen pressure of about 0.05 bar on the permeate side. The oxygen permeation fluxes increase with increasing temperature, pressure on the feed side, and oxygen concentration in the fed O-2-enriched air. An oxygen permeation flux of 10.2 cm(3) (STP) cm(-2) min(-1) was reached using O-2-enriched air with 50 vol % O-2 at 5 bar as feed and reduced pressure of 0.05 bar on the permeate side at 900 degrees C. A high oxygen purity up to 99.9 vol % and high permeation rates of almost 10 cm(3) (SIP) cm(-2) min(-1) were obtained at 900 degrees C in a 150 h operation under the conditions of 0.05 bar on the oxygen side and oxygen-enriched air at 4 bar on the feed side.