​High-silica (gel Si/Al = 100) SSZ-13 membranes were prepared by hydrothermal secondary growth on the surface of α-alumina hollow fiber supports. The membranes were evaluated for their performance in the separation of CO₂ from equimolar mixtures with CH₄ or N₂. The maximum CO₂–CH₄ and CO₂–N₂ separation selectivities were found to be 42 and 12 respectively, with a high CO₂ permeance of 3.0 × 10⁻⁷ mol m² s⁻¹ Pa⁻¹ at 293 K and total feed pressure of 0.6 MPa. At the low aluminum content, the prepared membranes contain a very low number of defects, as follows from their H₂/SF₆ ideal selectivity of over 500 in the 293–473 K temperature range. Due to their hydrophobicity, water in the feed mixture has only a small influence on the permeance at temperatures above 353 K. Water improves the CO₂–N₂ and CO₂–CH₄ selectivity, which is attributed to preferential blocking of the hydrophilic, non-zeolitic defect pores. The hydrothermal stability of the high-silica SSZ-13 membrane was evaluated by a long (220 h) CO₂–N₂ separation test with a humidified (9.5 kPa H₂O) feed mixture at 393 K and 0.6 MPa feed pressure. The permeance and selectivity were stable during this endurance test, underpinning the promise of high-silica SSZ-13 membranes for application in the separation of hot and humid gas mixtures.