The machine was one of only three in the world
Dennis Yates, 58, of No Man's Lane, Sandiacre, Derbyshire, was due to stand trial at Aylesbury Crown Court but changed his plea.
A separate charge of blackmailing Christine Large, the director of Bletchley Park, Buckinghamshire, where the machine was kept, was ordered to lie on file.
The case was adjourned for reports.
After leaving court, Yates said he had no comment to make but that he would be issuing a statement at a later date.
The machine, used by German intelligence during World War II to encrypt top-secret messages, went missing at an open day at Bletchley Park, Bucks, on 1 April last year.
Following the machine's theft a number of ransom demands were sent to Bletchley Park Trust, which runs the centre.
The device reappeared in October last year when it was sent to BBC Newsnight presenter Jeremy Paxman.
But three vital wheels from the machine were missing.
Police say they received a ransom note for £25,000 for the missing parts.
U-boat attacks
The Enigma team's work in deciphering codes - used by the Germans to direct operations including U-boat attacks on Allied convoys - proved vital to the outcome of the war.
Bletchley Park, code-named Station X, employed teams of mathematicians, linguists and chess champions during the war.
By the end of 1945, 10,000 people worked there.
With the help of decoding machines, the army of experts were able to crack the German code Enigma, which Berlin believed to be unbreakable.
The work carried out at the top-secret centre is believed to have shortened the war by several years and was kept secret until 1967.
The stolen device, an Abwehr Enigma G312, is a rare four-rotor version, one of only three still known to be in existence.
Bletchley Park director Christine Large said: "I think we have a result for all the people who want to see this machine back at Bletchley Park.
"It's a great tribute to the excellent work done by the police and people involved this case. They've done a fantastic job."