The flow of a fluid with a positive Joule-Thomson coefficient through a porous membrane is considered. Upstream of the membrane, the fluid is in the state of saturated vapor. Downstream of the membrane there is cooler, unsaturated vapor. Because a saturated vapor cannot simply cool down without condensation, fronts of phase change occur. The process is described assuming local thermodynamic equilibrium and an ideally wetting liquid phase. For a sufficiently small permeability of the membrane the fluid condenses fully at the upstream front of the membrane. Liquid flows through the membrane and evaporates completely at a front within the membrane. For very small permeabilities the fluid condenses upstreams and a liquid film forms in front of the membrane. Characteristic quantities are the critical permeability, $\kappa_\mathrm c = \nu_\mathrm lT_1(v_\mathrm g-v_\mathrm l) k_\mathrm l/\Delta h^2_\mathrm{lg}$, and a capillary number, $\mathrm{Ca}_\mathrm{JT}=\mu_\mathrm{JT}({\mathrm dp_\mathrm s}/{\mathrm dT})({\Delta p_{12}}/{\Delta p_\mathrm{cap,c}})$.