The problem of. reallocating the total endowment of an infinitely divisible commodity between agents with single-peaked preferences is considered. If debts (negative endowments and reallocations) are allowed, then an extension of the uniform allocation rule, the uniform reallocation rule, is shown to be the unique rule satisfying Pareto optimality, strategyproofness, and an equal-treatment condition. For the case that debts are not allowed, several variations on this result are presented.