So the British had intercepted (and partially decrypted) the Zimmermann Telegram, and knew they had a bomb on their hands, one whose explosion might hurry American entry into the war. The question was how to use it. Giving it to the Americans immediately would likely reveal to them that the British were reading American mail. That wouldn’t go over well.
In fact, Captain William “Blinker” Hall, the head of British decryption efforts, decided that even the British government itself couldn’t be trusted. Hall had a tendency to make fairly major policy decisions without consulting minor bodies like the Cabinet (including one time when Winston Churchill, of all people, remonstrated with him. Churchill remonstrating with someone for being too headstrong was like a rhinoceros crunching through a china shop and complaining about the mess the bull made). Here Hall decided that the Cabinet (including Prime Minister Lloyd George) could not be trusted not to reveal the secret to the Americans. Hall maneuvered things so that the telegram was given to the Americans without either them or the Cabinet realizing how the British had gotten it. 
Boghardt tells the story of what happened ably, and I won’t replicate it (yes, this is a subtle way to get you to buy the book). What I find interesting is that Boghardt points out that Hall continued the deception in the post-war era (most likely to allow the British to continue to listen to American transmissions), and it wasn’t until the first decade of this century that the deception was substantially undone. Hall told a false story in his (unpublished) memoirs and planted that myth with a historian, Burton Hendrick, writing a biography of the American ambassador in London. The story was picked up by Barbara Tuchman, who repeated it in her work, The Zimmermann Telegram.
You can use Google Scholar to trace the story, in a rough way. The Hendrick biography was cited quite a few times as was the Tuchman work. This is not to say that all those citations used or repeated the story about its interception, but it would be fascinating to trace those in which it did, and how the story spread. Alas, this is a blog post, and my time and space is limited.
And so a cover story remained the received history for almost a century, and there is left quite a lot of work to do on what it covered up, allies spying on allies. Thomas Boghardt was kind to send me a summary of this (never before seen historiographical analysis, only on Edge of the American West!):
While many works have been published on British cryptanalytic and espionage operations against the Germans in World War I and World War II, there is virtually nothing on the Allies spying on each other (with the notable exception of Soviet intelligence gathering on the West during World War II). The implications of inter-Allied espionage are potentially far-reaching if one considers, for example, that the British may have had advance knowledge of the U.S. delegation’s negotiating position in Paris in 1919, or Roosevelt’s willingness to confront Hitler and provide Lend Lease to the British in the early 1940s.
The narrative of history often gets smoothed out, with rough or unpleasant details shorn away. Events and people get described with a single adjective: the Good War, the Greatest Generation. The reality is deeper, more interesting, and often hidden.
 Drawn from Thomas Boghardt, The Zimmermann Telegram: Intelligence, Diplomacy, and America’s Entry Into World War I (Naval Institute Press, 2012), chapter 7.
 Private email correspondence, August 17, 2012. Used with permission.