The American Mathematical Society may or may not have a rule against it, but even if they don't I think this is a prohibitively bad idea.
At a conference, especially a big one like an AMS or AMS/MAA conference, time is the most valuable commodity: there are so many people there who would like to interact with each other and hear about each other's work, but so little time that many of the events take place on top of each other and the time given to each is very little. Notice that the standard length of the talks is 20 minutes. Virtually everyone who gives a 20 minute talk would rather give a longer talk (in particular, has much more to say). Most of the talks are kept
that short so as to involve the largest number of people in the very limited time available.
With this in mind, taking up two slots for the same talk may be viewed as remarkably selfish behavior. Many if not most people at these conferences are interested in more of the talks than they are able to attend: sometimes tough choices have to be made to see one talk versus another (some of us who attend these meetings remember "who we went up against" and marvel that anyone made it to our talk). To give the same talk twice really looks unfair and ungracious to the other participants. It would be hard for any session organizer to justify a "repeat performance" as a better intellectual outcome than a new talk.
I should say that at the JMM sometimes people give multiple talks. At a meeting a few years ago, I noticed that a certain hot job market candidate was giving three talks (of different kinds; maybe not more than one in a special session). If you're planning to give more than one talk in a special session I would definitely mention this to both session organizers: to have someone turned away while someone else speaks twice looks harsh and should be avoided most of the time, I think. This is a different thing though than giving the same talk twice: in fact, since giving two different talks is not unheard of in these meetings, giving the same talk twice looks perhaps even weirder: in particular, it may look like you very much want to talk but don't have very much to say.
Added: My colleague Brian Boe is the Associate Secretary for the Southeastern Section of the AMS. I consulted him, and he replied with the following information:
An individual may not give two talks on the same paper. (At JMM, this implies: not even in an AMS special session and an MAA invited paper session, for example).
An individual may not give more than one (10-minute) contributed paper talk.
An individual is allowed to speak in multiple special sessions (and up to one contributed paper session), as long as the talks are on different topics.