Correspondent Ed Bradley first interviewed Quasthoff in 1997, when he was about to give his first major performance at the Barbican Theater in London.
In spite of the applause that greeted him, he knows that when people see him for the first time, they are shocked by his appearance.
"One of my first concerts was in Braunschweig, a little city in the north of Germany. And we go in the church and the conductor come to my father, 'Oh, it's very nice that you sing here.' And my father, 'Sorry, it's not me. It's my son,'" says Quasthoff. "And the conductor looks down, really, like, 'Ooh!' and then I start to sing and he really loses his stick. It was really like, he was shocked."
Quasthoff was born near Hanover, Germany, in 1959. He was one of the first victims of the drug thalidomide, which his mother had taken during pregnancy to combat morning sickness. Unknown to doctors at that time, thalidomide had terrible side effects, causing physical deformities in the unborn child.
His voice, however, is exceptional by any standards. He can sing from high baritone to bass, and that big sound seems to come effortlessly from his small body.
"I accept my disability. It's a fact. I cannot hide it and I don't want to hide it, but I don't want to be judged as a disabled person," says Quasthoff. "I want to be judged as a singer, and I think my level is high enough that I have the right to be judged like this."
His parents, who thought he would never be able to lead a normal life, sent him to an institution for the disabled.
"My first sleeping room, I don't forget, was together with 10, 12 children. And they are not only physical handicapped, some are brain handicapped, also, so they cry in the night," recalls Quasthoff. "And it was a hard experience. But on the other side, now I say it was very good for me, because I really enjoy -- I try to enjoy every day, because I know how hard life really can be."
Quasthoff eventually returned home to live with his parents and enrolled in a regular school. He took up singing while his classmates played soccer in their free time. But when he tried to win a place at the local music school, he was turned down.
Students who wanted to study singing were also required to learn a musical instrument. And no matter how promising the voice, no exception could be made for a boy with no arms. So Quasthoff reluctantly decided to study law.
"It's very, very hard if you know and if you feel inside yourself, 'I am an artist and I want to be an artist,' and then to have this dry stuff," says Quasthoff. "It was really a hard time."
So Quasthoff took private singing lessons. And those lessons paid off. He's now considered by music critics to be one of the finest baritones of his generation, and gives about 80 concerts a year in Germany and around the world.
Does he have an entourage that follows him? "No, I'm sorry. Maybe there's a mistake," says Quasthoff. "I'm not Mick Jagger."
He has such an expressive face on the stage, and you can sense his joy when he's singing.
"I never trained in front of a mirror, you can be sure, never," says Quasthoff. "It comes out of myself."
Quasthoff often sings operatic arias like one from "Don Giovanni" in his concerts. But for obvious reasons, he has never performed in an opera. The only role he was offered was that of the dwarf in Verdi's "Rigoletto," and he turned it down.
"I think that my disability would be too much in the foreground. If you act, everybody looks, 'Oh, how he walks,'" says Quasthoff. "And I think it's very important to have this kind of singing intelligence to know exactly what is good for your voice, what is good for your career and what not."
He may not have the physique of an opera singer, but he admits that he sometimes has the temperament of one.
"If something goes wrong, sometimes I can really get furious," he says. "And I hate myself in this moment. And I'm working on it. I really want to be liked."
Quasthoff says he also wants to be loved, and he has an eye for a pretty woman. But he says what has so far prevented him from having a permanent relationship is not his disability, but the demands of his career.
"At the moment, I am in the situation where I said, 'It's better to be alone and happy than to have a relationship and have problems,'" says Quasthoff.
Quasthoff says he considers himself to have been born blessed to have made it to the top in a profession where he knows it helps to be good-looking, blessed with a loving family, and above all, blessed with the gift of music.
"It was a dream 10, 15 years ago, and now it's my normal life," says Quasthoff. "Sometimes it's crazy to realize."
But if he had to choose between being an able-bodied person who didn't have the ability to sing as well as he did, and a disabled person with his marvelous talent, what would be his choice?
"I'd stay like I am," says Quasthoff. "I would stay like I am."
What's changed since Bradley last met Quasthoff? Plenty. To begin with, he's finally achieved his dream – to perform in an opera, despite his concerns about how the audience might react to his disability.
But no one cared about his disability when he stepped onto the stage for the first time in 2003 to perform the role of Don Fernando in "Fidelio."
"It was a huge success, and for me, an unexpected success, because nobody spoke about, 'Is it really necessary that a disabled guy is singing now opera?'" says Quasthoff. "No, everybody really said, 'Of course, it's necessary. He has the voice, he has the talent.'"
And that talent over the last several years has brought Quasthoff even greater success. Still living in Germany, he's now in demand for other opera roles, and is performing with the most prestigious conductors and orchestras in the world.
In spite of his success, he still travels without an entourage. But, on this trip, when 60 Minutes Wednesday met with him again, he brought along his girlfriend, Claudia Schtelsick.
"Yes, I'm in love, which is wonderful," says Quasthoff. "She is intelligent, caring, and the best that happened in the last 45 years of my life, I would say."
Schtelsick, also from Germany, is a journalist who met Quasthoff almost a year ago, when she interviewed him for a local TV program. Recently, she accompanied him on a trip to Los Angeles, where he performed at the Walt Disney Concert Hall.
For some time, Quasthoff has not been content to sing just Mozart and Bach. At a small theater in Los Angeles, he showed his range as he stretched out with some old favorites. This is a side of Quasthoff his fans don't often see, and the audience couldn't get enough of it.
Everywhere we went with him, we heard that rich laugh. Life for Thomas Quasthoff has gotten as sweet as the songs he sings.
"My life is very, very fulfilled. And it's kind of an adventure at the moment," says Quasthoff. "Or I said sometimes a kind of film from which I don't know the end. I'm a very happy man."