AFAIK when the Pope declares something ex cathedra (‘out of his chair’) then it is seen as being infallible,a declaration of what the Church believes and that Catholics are to believe it as part of their dogma.
So what happens if the Pope decides to announce something in his official capacity that is at best rather spurious? He makes an official infallible declaration that Jesus did not exist or that Mary was not really the Mother of God?
Would it be considered to be part of Catholic dogma? Would the Catholic Church just ignore his declaration or decide that on this occasion he wasn’t quite that infallible?
If you’re Catholic, the answer is that he won’t, since he is indeed infallible when speaking ex cathedra.
If you’re not… haven’t a clue. Mighty interesting question. Why don’t you become Pope, hop in the chair and say Jesus never existed? Fighting ignorance and all that.
First of all, understand that this is extremely unlikely. For centuries now the College of Cardinals has always elected a pope from its own ranks (although there is no rule requiring that they do so). They know the guy well. Nobody who believes anything so contrary to Catholic doctrine, or who is likely to declare it anyway just to make a point, has a chance of getting so close to the papacy.
Add to this that statements made ex cathedra are extremely rare. The current Pope has never issued one, and I believe fewer than 10 such statements have ever been issued.
But ignoring all that… Technically, I suppose it would become doctrine if it was allowed to stand. However, if any pope ever did say such a thing and claim to be speaking ex cathedra, I believe the College of Cardinals would get together and declare that the Pope had taken leave of his senses and needed a long rest. A long looooong rest. Probably the rest of his life kind of rest. I believe you’d see a sort of quiet coup behind the scenes before the Pope was allowed to unilaterally contradict doctrine.
One of our knowledgedable Catholics could do better on this, but it’s worth remembering that for all the fuss about Papal Infallibility, it’s a pretty rare state of affairs when it’s invoked: only twice (Immaculate Conception and Assumption) to promulgate dogma, and fairly rarely to formally teach doctrine. And when it is, whatever is being promulgated has been discussed and analyzed extremely extensively in the Curia (Sacred Congregation on Dogma, IIRC). The Pope is the final decisionmaker and spokesperson, and is believed by Catholics to be guided by the Holy Spirit so that he will not promulgate error in this way, but it’s hardly a one-man show. (Do a parallel to the U.S. President – while Mr. Bush will announce American policy and has the final say on it, it’s a given thing that Messrs. Rumsfeld, Cheney, Rove, Powell, Ms. Rice, and a few others will have had a major hand in shaping that policy before it’s made public, and teams of intelligence people and Pentagon and State Dept. evaluators will have helped produce the estimates on which it depends.)
“Out of the chair” is probably a less-than-satisfactory translation – “emanating from the seat (of authority)” would probably be closer. It references the pronouncement of a formal teaching or decision, in any case – much as a judge may be informal in chambers but will place an air of formality in announcing a decision from the bench in open court.
And infallibility does not cover the everyday comments of a Pope; he’s a human being just like the rest of us, even in the official Catholic view. Before Parkinson’s took so strong a hold on him, JP2 was reputed to enjoy following football as a relaxation, and to “infallibly” pick the losing team to root for. 
Yep, he’s a huge football (“soccer” to us 'Merkins) fan. When he was young, he used to play, too–usually as goalkeeper.
Apparently, he likes to watch Liverpool play, though I hear that he really supports Lazio in Italy’s Serie A league. Liverpool’s been doing OK lately, but Lazio really could use some divine assistance (currently 16th in the league!).
Oh, and “ex cathedra” is better translated as “from the chair” instead of “out of the chair.” I always get this picture of the pope perched up on the Cathedra Petri when I hear the term, though I know he doesn’t do it that way.
There are two types of infallible teachings:
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A statement made ‘ex cathedra’ (‘from the seat’ i.e., from the office of the Bishop of Rome successor of Peter who was given the keys to bind and to loose from Christ who has been given all authority from the Father who needs no stinking deputation of authority). As noted above, this kind of infallible statement has only been made twice with regard to the Assumption of Mary and the Immaculate Conception of Mary.
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All the bishops of the world united together with the Pope say so, especially when in a Ecumenical Council.
Granted, it’s harder to pin down exactly what teachings fall into category #2, since the Ecumenical Councils didn’t have the forethought to say, “And now, something completely infallible…”
Certainly the statements of the Creed would fit as something that would be infallible under #2 (e.g., there is one God, etc…).
This would make null and void any crazy statement by the Pope (let’s say, JPII’s meds are off one day and at the balcony at St. Peter’s he says, “There is no God! - I infallible declare so from the seat of Peter!”). He would be reversing a teaching already understood to be infallibly taught, and he would certainly not enjoy communion with all the bishops of the world.
One might argue that since canon law provides no way to overrule the Pope, his new infallible teaching must stand. But, more than likely, the matter will be resolved as it usually is throughout history… someone close to the Pope forces him out of office and the next validly elected Pope redefines things… ‘more’ infallibly.
From the Catechism of the Catholic Church:
Peace.