So Ms. Fleming is trying to say goodbye on her own terms. “You don’t want people to be saying, ‘Oh my God, please stop,’” she said in London as she prepared to finish the “Rosenkavalier” run there. “Or, ‘I heard her when.’”

Her departure is a watershed moment for her extravagant, expensive art form, which is always imagining itself in trouble — what is opera about except crises? — but may really be in peril this time. Not only is opera more divorced than ever from mainstream culture, but also its core audience, the people who buy subscriptions, is literally dying off. The Met has had some luck attracting new operagoers through social media, collaborations with theater and visual artists, and fresher branding, but the most reliable way of ensuring attendance is still by casting big international stars, and one of Ms. Fleming’s magnitude is almost impossible to replace. Plácido Domingo, the only singer on her level still performing, is 76, and, though he keeps defying time, can’t go on forever; younger artists like Anna Netrebko and Jonas Kaufmann may be opera-famous, but are hardly household names.

“A superstar is that intangible thing,” said Mary Lou Falcone, the publicist who in 1998 helped guide Ms. Fleming through a crisis of confidence so severe she almost quit opera after being booed in Milan. “Nothing can explain it. After all the projections and trajectories, the public either latches on or it doesn’t.”

Drawn to Ms. Fleming’s combination of glamour and accessibility — she became known as “the people’s diva” — the public did latch on. Invited to sing David Letterman’s Top 10 list, and to record, in the original Elvish, some of the soundtrack for the third “Lord of the Rings” movie, she gained a following among people who, strictly speaking, weren’t opera buffs at all. In many cases, Ms. Fleming was the first and only opera singer they’d ever paid attention to. She has sold over two million records, a huge number for opera, and won four Grammy Awards. In 2014, she became the first opera singer to deliver the national anthem at the Super Bowl. In 2015, she made the jump from Lincoln Center to Broadway to appear (as a tetchy diva) in the play “Living on Love.”