The artifact was stolen late Friday from the astronaut's namesake museum in his small hometown of Wapakoneta.
Local police are working with only a blurry image of the thief taken via security camera, which has not helped them identify the suspect.
The five-inch tall solid gold model of the 1969 Lunar Excursion Module, which took the first humans to the Moon, was one of three made by Cartier.
Authorities were unable to give an estimated value for the object, but the museum said the loss could not be measured in dollars and cents.
"Theft from a museum is a theft from all of us," the Armstrong Air & Space Museum said in a Facebook post.
"For every day that an item is missing, we are all robbed of an opportunity to enjoy it and our history."
Along with the gold replica, other items in the same exhibit display were also taken, including award medals and presentation coins, police said.
The museum employed video surveillance, door alarms and other security measures, which it would not specify. Authorities did not say how the thief managed to best those measures.
"This is an ongoing criminal investigation," said Wapakoneta Police Chief Russel Hunlock in a statement.
"I will continue to release information as I can, but I will not tarnish the integrity of the investigation."
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