The Answer:

Contrary to popular belief, the U.S. Postal Service has no "official motto."

The familar sentence you are thinking of is this:

Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds.

This is commonly misidentified as the creed of our mail carriers, but actually it is just the inscription found on the General Post Office in New York City at 8th Avenue and 33rd Street.

Here's how the official Web site of the U.S. Postal Service describes the origin of the inscription.

This inscription was supplied by William Mitchell Kendall of the firm of McKim, Mead & White, the architects who designed the New York General Post Office. Kendall said the sentence appears in the works of Herodotus and describes the expedition of the Greeks against the Persians under Cyrus, about 500 B.C. The Persians operated a system of mounted postal couriers, and the sentence describes the fidelity with which their work was done. Professor George H. Palmer of Harvard University supplied the translation, which he considered the most poetical of about seven translations from the Greek.

—The Fact Monster