MOSCOW—The director of Russia’s military intelligence agency has died unexpectedly, according to a short statement released Monday on the Kremlin website, which didn't specify the cause of his death. Col. Gen. Igor Sergun had run the Main Intelligence Directorate of Russia’s General Staff, known as the GRU, since late 2011. He was 58 years old. Russian Defense Minister Sergei Shoigu, in a statement released to the Interfax news agency, said Col. Gen. Sergun died suddenly on Sunday. Mr. Shoigu’s statement offered no additional details.