IPhone 4s (iOS 6) keeps asking for my Apple ID and password when I open Facebook?

I just got my iPhone for Christmas, and I have been able to use Facebook before perfectly fine but now all of a sudden I get a message that says "Apple ID Password" and has a space for "username" and for "password". I type both in correct, but the bubble closes and goes to the home screen. Then, I open Facebook again, and the process repeats.

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