Clutching a diamond engagement ring, Warren Buffett got down on one knee and asked for Alexa Tavasci's hand in marriage.

"Please take me. Please have me," the 81-year-old billionaire begged at an Omaha, Neb., restaurant as cameras snapped around him.

Ms. Tavasci, a 21-year-old junior at Northern Arizona University in Flagstaff, accepted the...