
Washington circa 1925. "White House Zero Mile." National Photo Company Collection glass negative, Library of Congress. View full size.
The milestones must have been in vogue in the 20's. A very similarly inscribed one can be found in front of the MacArthur Memorial in Norfolk, VA.
Lots of good information in this post, but I am left with the simple query, "is it still there?"
[Google "zero milestone." - Dave]

That's a very odd cloud formation. Is it a flaw on the photograph? If it is, why aren't there any white dots on the tree branches?
[That's mold growing on the emulsion. There's no emulsion (or very little) on the darkest parts of the image, hence no spots there. - Dave]
Oh, I long for the day when our country recognized the importance of infrastructure and celebrated improvements. I particularly like Harding's statement concerning highways as the "arteries of the enlightened state." The basic facts regarding the zero milestone are on Wikipedia.
60 Trucks Will Cross Continent
Monument Will be placed on Ellipse Tomorrow to Mark Start.
Ceremonies on the north side of the Ellipse, between the monument and the white house, will be held at 10 o'clock tomorrow morning to mark the start for a motor convoy of 60 trucks and other vehicles for San Francisco, the first transcontinental trip of its kind ever undertaken.
Secretary of War Baker will deliver and address and a small monument will be erected at the starting point as the zero milestone of the national highway system. The monument is designed to be the milestone from which, as from the golden milestone in the Forum of Rome, all road distances will be reckoned.
A committee representing the American Automobile Association, cooperating with the motor transport corps of the army, is in charge of arrangements. Dr. S.M. Johnson is chairman.
President Accepts Zero Stone Today
Nobles of the Mystic Shrine, tourists and Washingtonians will witness the dedication of the Zero Milestone monument immediately south of the White House this afternoon at 2 o'clock. President Harding will make the address of acceptance and dedicate the monument, which will be presented by Dr. S.M. Johnson, general director of the Lee Highway association.
Text of President's Address
My countrymen, in the old Roman forum there was erected in the days of Rome's greatness, a golden milestone. From it was measured and marked the system of highways which grid-ironed the Roman world and bound the outermost provinces to the heart and center of the empire.
We are dedicating here another golden milestone, to which we and those after us will relate the wide-ranging units of the highway system of this country. From the golden zero stone of the Roman highway system the soldiers, the adventurers, the administrators and later the missionaries of Rome went out to carry the institutions and the civilization of the imperial state to their expanding world. The Romans were the highway builders of the ancient world, and we shall find inspiration on this occasion in the thought of how greatly their highways influenced the course of all history since their time. Along these ancient highways surged the great conflicting forces of civilization and barbarism - now ebbing, now flowing, but always influencing the barbarians toward the civilization, the institutions, the religion of the Italian peninsula. Indeed, it would require little imagination to picture our civilization in terms of distance, by miles or centuries, from that ancient Roman community which first conceived the highways as the arteries of the enlightened state. Always the avenues of trade were the paths of widening civilization.
... [10 more paragraphs]....