They have eavesdropped on the enemy for decades, tracking messages from Hitler’s high command and the Soviet K.G.B. and on to the murky, modern world of satellites and cyberspace. But a lowly and yet mysterious carrier pigeon may have them baffled.

Britain’s code-breakers acknowledged Friday that an encrypted handwritten message from World War II, found on the leg of a long-dead carrier pigeon in a household chimney in southern England, has thwarted all their efforts to decode it since it was sent to them last month.

As the bird’s story made headlines, pigeon specialists said they believed it may have been flying home from British units in France around the time of the D-Day landing in 1944 when it somehow expired in the chimney at the 17th-century home where it was found in the village of Bletchingley, south of London.

After sustained pressure from pigeon fanciers, Britain’s Government Communications Headquarters, its code-breaking and communications interception unit in Gloucestershire, agreed to try to crack the code. But on Friday the secretive organization acknowledged that it had been unable to do so.