DUBAI—Saudi authorities on Saturday released billionaire Prince al-Waleed bin Talal, more than two months after he was detained in a widespread purge of the kingdom’s elite.

Saudi officials said Saturday that Prince al-Waleed came to an undisclosed settlement with the government that allows him to remain chairman of his company, Kingdom Holding Co., one of the country’s biggest conglomerates. He is also a major investor around the world, including large stakes in Twitter Inc. and the Four Seasons hotel chain. The status of...