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Slightly more than 200 cards. I think 207, to be exact, although I don't have the game with me to check. If you have a copy of the game, you can go in the card listing (not the one where you customize your decks, but the one off the main menu that lists every card or the back of the card if you haven't acquired one yet), then count the number of card backs in each row and how many columns it has, you can multiply them to find the exact number.
And each card is pretty unique, also. There's also no duplicates that I'm aware of.