"I read a book during the meal."
Translation:Leggo un libro durante il pasto.
Because mentre is a conjunktion and durante is a preposition. And conjunktion - as far as I know - combine two sentences. So (and that's more like a guess) I would say, if you want to use mentre here, the sentence would have to be: "Leggo un libro mentre mangio il pasto" - "I read a book while I eat". I'm not native English but I just learnt, that it's the same in english. During is a prep and while is a conj.