A membrane assembly for the permeative separation of a fluid from fluid mixtures includes a porous, fluid-permeable, metallic support substrate, a membrane that is disposed on the support substrate and is selectively permeable to the fluid to be separated off, and a connecting part which is formed, at least on the surface, of a fluid-tight, metallic material. The support substrate is cohesively bonded along a peripheral section thereof to the connecting part. A ceramic, fluid-permeable, porous, first intermediate layer is disposed between the support substrate and the membrane. At least one ceramic bonding layer is disposed directly on the connecting part and the material bond and extends at least over the cohesive material bond and an adjoining section of the connecting part. The first intermediate layer ends on the bonding layer and has a greater average pore size than the bonding layer.