We consider the problem of reallocating the total endowment of an infinitely divisible commodity among agents with single-peaked preferences and study several properties of reallocation rules such as individual rationality, endowment monotonicity, no-envy, and bilateral consistency. Our main result is the proof that individual rationality and endowment monotonicity imply Pareto optimality. This result is used to provide two characterizations of the uniform reallocation rule. The first characterization states that the uniform reallocation rule is the unique reallocation rule satisfying individual rationality, endowment monotonicity, and no-envy. In the second characterization, no-envy is replaced by bilateral consistency.