Scalia says he gave the priesthood some thought.
"And decided no?" Stahl asks.
"And decided he was not calling me," Scalia explains.
"What is the connection between your Catholicism, your Jesuit education, and your judicial philosophy?" Stahl asks.
"It has nothing to do with how I decide cases," Scalia replies. "My job is to interpret the Constitution accurately. And indeed, there are anti-abortion people who think that the constitution requires a state to prohibit abortion. They say that the Equal Protection Clause requires that you treat a helpless human being that's still in the womb the way you treat other human beings. I think that's wrong. I think when the Constitution says that persons are entitled to equal protection of the laws, I think it clearly means walking-around persons,"
Appointed by Ronald Reagan, he was sworn in at age 50, the first Italian-American to ever serve on the Supreme Court.
He met his wife Maureen in Cambridge when she was a senior at Radcliffe and he was in his last year at Harvard Law School. They have been married for 48 years and rarely disagree, they say.
"She says she could have married so-and-so...," Scalia says.
"Oh, not really," his wife replies.
"You do. You say that. And of course the reason she didn't was that 'so-and-so [was] wishy-washy,'" Scalia says.
"This is absolutely true. He will say, 'You would have been bored.' I say, 'Oh, that's right!' I would have been bored," Maureen Scalia says. "I would have been bored."
But she says she hasn't been bored.
"Whatever my faults are, I am not wishy-washy," Antonin Scalia adds.
The marriage has flourished: they have 9 children and 28 grandchildren.
Why so many children?
"Well, as someone said, they're both overachievers, I guess," Maureen Scalia jokes, laughing.
"Well, we didn't set out to have nine children. We're just old-fashioned Catholics, you know? Playing what used to be known as 'Vatican roulette,'" Scalia jokes.
The Scalia children, ranging in age from 27 to 46, are all conservative, all successful, including two lawyers, a major in the Army, a poet, and a priest.
"If in an old-fashioned Catholic family with five sons you don't get one priest out of it, we're in big trouble, right?" Scalia jokes. "I will say that the other four were very happy when Paul announced that he was going to take one for the team. I don't know."
"The justice told us that he didn't go to the soccer games and the piano recitals and things," Stahl tells Maureen Scalia.
"You know, my parents never did it for me," Antonin Scalia interrupts. "And I didn't take it personally. 'Oh Daddy, come to my softball game.' No, I mean, it's my softball game. He has his work. I got my softball game. Of course, she was very loyal. She went to all the games."
"Most," Maureen Scalia adds. "I would get five minutes at each on a Saturday."
All their children are grown up now. And Scalia, after 22 years on the court is starting another career as an author. His new book, "Making Your Case, The Art Of Persuading Judges," is surprisingly breezy in that it's a primer for lawyers on how to win cases. His co-author is Bryan Garner, an expert on legal writing.
"You say things in it like, 'Be prepared. Look the judge in the eye.' You almost make it sound like lawyers are imbeciles," Stahl says.
"You would be surprised," Scalia replies, laughing.
They wrote the book together, occasionally sitting side by side, arguing. Surprisingly, Garner says, it was the justice who often showed humility by yielding.
"I thought you punched pretty hard. You threw me a hard punch. And then sometimes he'd just want to see: could I punch back on the counterpoint. But often he could be brought around. He could be persuaded," Garner explains.
"That doesn't show that I'm humble. It just shows that I'm not stupid," Scalia says.
"I thought you were very deferential, and surprisingly so. It was disarming to me." Garner adds.
Scalia deferential? That's something you never hear about him on the court, where he has been unable to persuade his fellow justices to come over to his way of thinking. The only other originalist on the court is Justice Clarence Thomas.
"A lot of people thought that when you joined the court you would use your charm to bring the other justices around to side with you. And it hasn't happened," Stahl says.
"I'm not going to change their basic philosophy. These people have been thinking about the law for years. They're not going to suddenly say, 'Oh God, Nino, explain it all to me.' I understand that's not going to happen," Scalia says.
That awareness has at times brought this man - usually so confident and charge ahead - to bouts of despair, and he was taken aback that we knew about it.
"You've apparently had some down times in your tenure on the court so far. And I'm pointing to the term of 1995-96 when you wrote to former Justice Blackmun at the time, and here's what you said: 'I am more discouraged than I have been at the end of any of my previous nine terms.' You also wrote that you were beginning to repeat yourself, and you did not see much 'use in it anymore,'" Stahl remarks.
"Gee, I hadn't remembered that I'd written it," Scalia says.
"It says, 'I am beginning to repeat myself,'" Stahl says.
"That's true. That is something that gives me some concern. I mean after a while, you know, I'm saying the same things in today's dissent that I said in a dissent 20 years ago," Scalia explains.
"Around that same time you wrote, 'The court must be living in another world. Day by day, case by case it is busy designing a Constitution for a country I don't recognize,'" Stahl says.
"Yeah. That's how I felt," Scalia says.
"Past?" Stahl asks.
"It's been less dire in more recent years," Scalia replies.
"In other words, you've had down times," Stahl asks.
"Yeah, I think so. I'm happier sometimes than at other times. And the end of a term, I don't care what term it is, it's usually a disappointment," Scalia says.
That's because - until recently - he was often on the losing side in cases he cared about most. Over the last several years Scalia has reached outside the court, speaking out publicly about his philosophy, in hopes of influencing the next generation. It's a role he relishes.
"Little kids come to the court, they're brought by their teachers. And they recite very proudly what they've been taught. I mean, this is how widespread the no-'The Constitution is a living document.' And I have to tell them 'It's a dead document,'" Scalia told the students at the Oxford Union.
He says the speeches energize him, but at 72, Stahl wondered if he ever thinks about retiring.
"When I first came on the court I thought I would for sure get off as soon as I could which would have been when I turned 65. Because you know, justices retire at full salary. So there's no reason not to leave and go off and do something else. So you know, essentially I've been working for free, which probably means I'm too stupid to be on the Supreme Court," Scalia says, laughing. "You should get somebody with more sense. But I cannot - what happened is, simply I cannot think of what I would do for an encore. I can't think of any other job that I would find as interesting and as satisfying."