WASHINGTON—The U.S. Supreme Court allowed President Donald Trump’s administration to implement part of his temporary ban on travelers from six Muslim-majority countries and said Monday it would give full consideration to whether the president’s actions were lawful.

The Supreme Court’s action, in a case of unusually high stakes for a new president, is a significant reversal of fortune for Mr. Trump, who had been on the losing end of several lower-court decisions that blocked his March 6 executive order.

That...