This paper proposes a new mechanism for combinatorial assignment—for example, assigning schedules of courses to students—based on an approximation to competitive equilibrium from equal incomes (CEEI) in which incomes are unequal but arbitrarily close together. The main technical result is an existence theorem for approximate CEEI. The mechanism is approximately efficient, satisfies two new criteria of outcome fairness, and is strategyproof in large markets. Its performance is explored on real data, and it is compared to alternatives from theory and practice: all other known mechanisms are either unfair ex post or manipulable even in large markets, and most are both manipulable and unfair.