While a government agency oversees the excavations, the dig and the site are largely funded and run by something called "El'Ad." Spielman works for El'Ad, which claims they're not a settlers' organization, though people we spoke to say they are.
"I think that it would be correct to call us an organization who believes deeply in the history of Jerusalem," Spielman said.
Asked if it's all archeology, he told Stahl, "Archaeology and rebuilding a Jewish neighborhood."
"So El'Ad is doing archaeology and settlements?" Stahl asked.
"We are doing archaeology and we are buying homes and buying land," he replied.
"But is it El'Ad's goal to ease the Arabs away from right where we are right now?" Stahl asked.
"Put it this way, if there's a home that an Arab wants to sell and I have the money to buy it, and I can move, enable a Jewish family to live there, and I can dig archaeologically underneath it, then I think that's a wonderful thing to do," he said.
The Arabs say it's a provocative thing to do. Devout Jews Yonatan and Devorah Adler live in one of the houses El'Ad bought. El'Ad has raised tens of millions of dollars, half from the United States, and buys the homes on land the Palestinians claim for a future state.
The Adlers raise their six kids on the actual dig. "The City of David is where Jerusalem began," Yonatan Adler said. "This is where prophets walked. This is where half of the Bible was written. This is what we're talking about."
"And yet, when you see those maps, it's over in the Palestinian side," Stahl pointed out.
"Yeah, well, maps are written on paper. This is written on our hearts," he replied.
"But it is one of the proposals on the table," Stahl said.
"I'll tell you. Jerusalem cannot be divided. Jews would never allow it," Adler said.
The Adlers say they don't mind living behind gates, and having guards on their roof. They would never consider leaving.
"But you're like a soldier on the front line," Stahl remarked.
"I don't think we see ourselves as soldiers at all. We see ourselves as everyday regular people, living in a very, very special place," Devorah Adler replied.
Palestinian Jawad Siyam was born in this "very, very special place" and says he can trace his roots there back 930 years. He's pessimistic about the Palestinians ever having their own state.
"What will happen to this village if there's a two-state solution?" Stahl asked.
"I don't think there will be a two-state solution. That's not possible to do it. Today they, the settler groups, are much stronger than before. The settler groups in Jerusalem, they are controlling," he said.
He's angry that El'Ad bought his grandmother's house and moved a number of Jewish families in, so he's become an activist leader in Silwan, where there has been a string of escalating confrontations.
In one protest we attended, at the City of David, Jawad got into a shouting match with a site-worker behind a gate. "You will be the rubbish of the history!" he shouted. "There is no proof that King David was here. You want to take our land."
Some of the incidents have become violent, like one on October 8. Boys were throwing stones at passing cars; two of the boys got hit by a car.
The driver was, of all people, the head of El'Ad. He later went to the police and said he hadn't stopped because he felt his life was in danger. Both boys survived, one with a broken leg.
Fighting often erupts there with Israeli guards brought in to protect the settlers. "Clashes are daily inside Silwan between the villagers and settlers and the gun guards for the security there," Jawad told Stahl.
"The government pays for the gun guards?" she asked.
"It's tax money. It's, I pay it. Everyone who is paying taxes is paying it," he replied.
"You pay taxes and that money goes to pay for the guards to guard the settlers," she remarked.
"Yes, of course," Jawad said.
"So you're helping guard the settlers," Stahl remarked.
"Yeah, I'm a fan of the settlers and the gun guards," he replied sarcastically.