"A lot of positions I take now, no one would've thought I would've taken. Who would have thought 20 years ago I'd be leading a march for immigration? Or that I would support same-sex marriage, which most black church people don't. So I think that a lot of people are stuck in time. Thankfully, I'm not," Sharpton said.
The new Sharpton has evolved in some ways that surprised us. For instance, one of his favorite haunts is the Havana Room, a fancy private midtown cigar club, whose other members are mostly white Wall Street bankers and the city's elite.
"You're a civil rights leader, representing all that that suggests, implies. And then there's this other side of you. You live in the Upper East Side, fancy place. You come here, fancy place," Stahl pointed out.
"I don't see that as a contradiction," he replied. "This idea that we have to only be in one area: we fought for access. So why wouldn't we use access? Civil rights has nothing to do just with poor people. Civil rights, everybody has a problem. Racial profiling, one of the things that I became known for, came to me from people who had huge multi-million homes in New Jersey, 'cause blacks were being stopped in their Mercedes-Benz," Sharpton said.
He smokes cigars at the club during the week; on many weekends, he's preaching.
He doesn't have his own congregation, so like Martin Luther King Jr, he told us he preaches at a different black church almost every Sunday.
Sharpton actually started preaching when he was just four. He was raised in a middle class section of Brooklyn. When he was 10 his life was turned upside down when his father abandoned the family.
Sharpton's dad fathered a child with his half sister. "My mother's child," he told Stahl.
When his father left, he and his mother were forced onto welfare.
"We became a laughing stock. Every kid on the block used to point at us, laughing. 'Their lights are out, his father's gone. They repossessed the Cadillac.' It was a haunting experience. And, it made me have a lifelong search for fathers," Sharpton remembered.
And for attention, he said. He made his mark as a civil rights leader in New York in the 1980s by leading angry marches. They led to change, but one nearly caused a race riot.
"People have said that you're a hate monger, a racial ambulance chaser, a shakedown artist, a race baiter. Did you go too far?" Stahl asked.
"No," Sharpton replied.
"You don't regret anything you said back then?" she asked.
"If you want to say that I use language sometimes that I shouldn't, yes. If you want to say that I had more vanity than I should've, yes. But don't say that I was a hater and violent. 'Cause I was never that," Sharpton said.
When Sharpton was in his 20s and 30s, there were allegations of mob ties, never proved; and charges he used his civil rights organization to shake down businesses for contributions, which he denies.
"I think he has been a hustler all of his career," Wayne Barrett, an investigative reporter with The Daily Beast, told Stahl.
Barrett has written about Sharpton for more than 20 years. "I think he's in the civil rights business, I don't think he's a civil rights leader. I think he's in the business. He has an organization called the National Action Network; nobody knows what has happens to all that money," he told Stahl.