A carbon molecular sieve (CMS) membrane having improved separation characteristics for separating olefins from their corresponding paraffins is comprised of carbon with at most trace amounts of sulfur and a group 13 metal. The CMS membrane may be made by pyrolyzing a precursor polymer devoid of sulfur in which the precursor polymer has had a group 13 metal incorporated into it, wherein the metal is in a reduced state. The pyrolyzing for the precursor having the group 13 metal incorporated into it is performed in a nonoxidizing atmosphere and at a heating rate and temperature such that the metal in a reduced state (e.g., covalently bonded to carbon or nitrogen or in the metal state).