...the flesh is weak" comes out as "The meat is rotten, but the wine's great".

Seems like I have to repeat myself over and over again, since this is an urban legend. According to MACHINE TRANSLATION: An Introductory Guide:

The `spirit is willing' story is amusing, and it really is a pity that it is not true. However, like most MT `howlers' it is a fabrication. In fact, for the most part, they were in circulation long before any MT system could have produced them (variants of the `spirit is willing' example can be found in the American press as early as 1956, but sadly, there does not seem to have been an MT system in America which could translate from English into Russian until much more recently --- for sound strategic reasons, work in the USA had concentrated on the translation of Russian into English, not the other way round). Of course, there are real MT howlers. Two of the nicest are the translation of French avocat (`advocate', `lawyer' or `barrister') as avocado, and the translation of Les soldats sont dans le café as The soldiers are in the coffee. However, they are not as easy to find as the reader might think, and they certainly do not show that MT is useless.

BTW, since this book is no longer available in the stores, the whole contents is placed online. I recommend reading this book to anyone who is interested into the subject of MT. It really is a nice introduction into the subject.