We consider the problem of allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. A rule that has played a central role in the analysis of the problem is the so-called uniform rule. Chun (200 1) proves that the uniform rule is the only rule satisfying Pareto optimality, no-envy, separability, and Q-continuity. We obtain an alternative characterization by using a weak replication-invariance condition, called duplication-invariance, instead of Q-continuity. Furthermore, we prove that the equal division lower bound and separability imply no-envy. Using this result, we strengthen one of Chun's (2001) characterizations of the uniform rule by showing that the uniform rule is the only rule satisfying Pareto optimality, the equal division lower bound, separability, and either Q-continuity or duplication-invariance.