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We determine the reliability function of the exponential-server timing channel (ESTC) in the limit as the data rate approaches zero. The limit shows that at low rates, the ESTC is strictly more reliable than the Poisson channel without dark current, answering a question Arikan posed in these Transactions. The proof employs a distance metric over inputs to timing channels that parallels Euclidean and Hamming distance for conventional channels. A consequence of the proof is that bounded-distance decoding, with distance measured according to this metric, is exponentially optimum for the ESTC in the low-rate regime. We also prove the straight-line bound for the channel and a bound on the reliability of timing channels with general service distributions in the limit as the data rate approaches zero.