Misty Copeland was fast becoming the most famous ballerina in the United States — making the cover of Time magazine, being profiled by “60 Minutes,” growing into a social media sensation and dancing ballet’s biggest roles on some of its grandest stages. But another role eluded her: She was still not a principal dancer.

Until Tuesday, when Ms. Copeland became the first African-American woman to be named a principal in the 75-year history of American Ballet Theater.