Nowhere does it say that the Mouth of Sauron was immortal. All that we know is that he prolonged his life with sorcery. You can still be very long-lived and not be immortal.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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