WASHINGTON—Stephanie Grisham, the first lady’s communications director, will serve as the next White House press secretary and communications director, Melania Trump said Tuesday.

Mrs. Trump praised Ms. Grisham in a tweet. “She has been with us since 2015 - @potus & I can think of no better person to serve the Administration & our country. Excited to have Stephanie working for both sides of the @WhiteHouse.”

President...