“Then King Darius made a decree. They searched the archives where the documents were stored in Babylon. However, it was in Ecbatana, the capital which is in the province of Media, that a scroll was found on which this was written.”
King Darius decreed that a search should be made in the archives at Babylon for the scroll of King Cyrus. Guess what? Very conveniently, King Darius found the scroll from King Cyrus in Ecbatana in Media, which was the summer home of the king. Media was on the Caspian Sea and had become part of Persia, after being independent before that. Also King Darius had a residence there. Why it was there is not clear, except that King Cyrus may have issued it from there.