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The general elliptic fiber is shown to be a complete intersection of two non-generic quadrics in P^3 and resolved elliptic fibrations are obtained by embedding the fiber as the generic Calabi-Yau complete intersection into Bl_3 P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitely many Calabi-Yau elliptic fibrations. 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