Looks like this phrase, which is not an official motto of the U.S. Postal Service, comes from Herodotus, describing the Ancient Persian Postal Service. Wikipedia:
“The building prominently bears the inscription: Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds, which is frequently mistaken as the official motto of the United States Postal Service. It was actually supplied by William Mitchell Kendall of the firm of McKim, Mead & White […] The sentence is taken from Herodotus’ Histories (Book 8, Ch. 98) and describes the faithful service of the Persian system of mounted postal messengers under Xerxes I of Persia. The U.S.P.S. does not actually have an official motto or creed, but nonetheless the inscription on the building is often cited as such.”
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