This is often mistaken for an official motto of the Unites States Postal Service. It is merely what the arhitect added to the 1913 building. Some have claimed that the Herodotus citation doesn't quite read this way.
The building itself is scheduled to become part of Penn Station in 2009; the building will be renamed after the late New York senator Patrick Moynihan. It is unknown if the motto will remain in place when the post office departs.
But you know how these things work in New York. Madison Square Garden is no longer at Madison Square, Manhattan College is in the Bronx, and egg cream has neither egg nor cream
28 December 1913, New York Times, pg. 12:
These support a cornice, on which is engraved the motto which Herodotus wrote in tribute to the couriers who dashed through Persia with the mails and military orders: "Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds."
6 August 1925, Bridgeport (CT) Telegram, pg. 22, col. 2:
Consider the slogan of the Service:
"Neither snow nor rain
Nor heat nor gloom of night
Stays these couriers
From the swift completion
Of their appointed rounds."
25 December 1929, New York Times, pg. 3:
Despite unfavorable weather and a volume of mail which surpassed that of previous years by more than 1,400,000 pieces, Postmaster John J. Kiely and his aids have handled the holiday rush, the records show, in unerring emulation of the passage from Herodotus which adorns the facade of the General Postoffice at Eighth Avenue and Thirty-second Street.
One of the most conspicuous demonstrations of the fact that "neither snow, nor rain, nor gloom of night" stays the couriers of the Postoffice Department from the "swift completion of their appointed rounds," according to Assistant Postmaster Charles Lubin, was the distribution yesterday of more than 6,000 sacks of Christmas mail which arrived from Europe on the Leviathan.