Mr Trump entered a small elevator taking him to the third floor of the Apostolic Palace and, after a long ceremonial walk past frescoed corridors, shook the pope's hand at the entrance to the private study, which the frugal pope uses only for official occasions.
Francis smiled faintly as he greeted Mr Trump outside the study and was not as gregarious as he sometimes is with visiting heads of state. Mr Trump, seeming subdued, said "it is a great honour."
Even when the two were sitting at the pope's desk in the presence of photographers and reporters, the pope avoided the kind of small talk that usually occurs before the media is ushered out.
The two talked privately for about 30 minutes with translators.
Both men looked far more relaxed at the end of the private meeting, with the pope smiling and joking with Mr Trump and his wife Melania.
Francis gave the president a small sculptured olive tree and told him through the interpreter that it symbolised peace.
"It is my desire that you become an olive tree to construct peace," the Pope said, speaking in Spanish through an interpreter.
Mr Trump responded: "We can use peace."
Francis also gave Mr Trump a signed copy of his 2017 peace message whose title is "Nonviolence - A Style of Politics for Peace," and a copy of his 2015 encyclical letter on the need to protect the environment from the effects of climate change.
"Well, I'll be reading them," Mr Trump said.
Mr Trump's softer stance on environmental regulations is at odds with Francis' view that climate change is caused mostly by human activity.
Mr Trump gave the pope a boxed set of five first edition books by slain U.S. civil rights leader Martin Luther King.
As Mr Trump and the pope said goodbye at the door of the study, Mr Trump told the pope: "Thank you, thank you. I won't forget what you said."
The meeting with the pope was the third stop on a nine-day foreign tour due to end on Saturday, and part of his world tour of religions after meeting leaders of Muslim nations in Saudi Arabia and visiting holy sites in Jerusalem.
While his talks in Saudi Arabia and Israel were mostly friendly, the meeting between the head of the Roman Catholic Church and the thrice-married, blunt-spoken Mr Trump had the potential to be a little more confrontational.
The pope said last year a man who thinks about building walls and not bridges is "not Christian," a sharp reprimand for Mr Trump's vow to build a wall along the U.S. border with Mexico.
Mr Trump said it was "disgraceful" of the Argentine-born pope, who represents just over half of the world's two billion Christians, to question his faith.
"If and when the Vatican is attacked by Isis, which as everyone knows is Isis' ultimate trophy, I can promise you that the pope would have only wished and prayed that Donald Trump would have been president," Mr Trump said during the campaign.
The Vatican also took a dim view of Mr Trump's anti-Muslim campaign rhetoric, although he softened his tone considerably in a major speech in Riyadh.
Part of Mr Trump's motivation for meeting the pope was to dramatise how the three major religions should rally against the threat from Islamist militants.
Mr Trump at first did not plan to stop in Rome during his visit to Europe, which some in the Vatican saw as a snub. When he changed his mind, the Vatican squeezed him in at 8.30am on a Wednesday morning, an unusual day and an unusually early time.
After the meeting, Francis held his weekly audience with the general public in St Peter's Square.
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