Complex sentence: On May 31, after 11 days of secret detention, the US State Department announced that he had been fired and "detained by order of the Home Secretary", but not that he had been arrested under the Official Secrets Act.
Split into simpler sentences: On May 31, after 11 days of secret detention, the US State Department announced that he had been fired and "detained by order of the Home Secretary". The statement did not say that he had been arrested under the Official Secrets Act.