Under clear blue skies, Trump, who exchanged sharp words with the pope during the U.S. election campaign last year, received a tribute from the Swiss Guard in a Vatican courtyard when he arrived.
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 pontiff uses only for official occasions.
Before the door of the wood-lined elevator closed, a Vatican protocol official was heard quipping to the president that it was not "like Trump Tower in New York".
Francis smiled faintly as he greeted Trump outside the study and was not as outgoing as he sometimes is with visiting heads of state. 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 Trump and his wife Melania.
Francis's interpreter could be heard translating a comment by the pope to the First Lady: "What do you give him to eat?".
Francis then 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.
Trump responded: "We can use peace."
Francis also gave 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," Trump said.
Trump's softer stance on environmental regulations is at odds with Francis' view that climate change is caused mostly by human activity.
Trump gave the pope a boxed set of five first-edition books by murdered U.S. civil rights leader Martin Luther King.
As Trump and the pope said goodbye at the door of the study, Trump told the pope: "Thank you, thank you. I won't forget what you said." Trump met separately with top Vatican diplomats and visited the Sistine Chapel before leaving the Vatican.
A Vatican statement said the meeting was "cordial" and that the Vatican hoped there could be "serene collaboration" between the U.S. government and the American Catholic Church, including "assistance to immigrants".
The U.S. Catholic Church hierarchy opposes Trump's attempt to cut federal assistance for cities that give sanctuary to illegal immigrants.
It also opposes his plan to build a wall on the U.S. border with Mexico. The pope said last year a man who thinks about building walls and not bridges is "not Christian".
Trump, who was a candidate at the time, responded that it was "disgraceful" of the Argentine-born pope, who represents just over half of the world's two billion Christians, to question his faith.
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 Trump had the potential to be a little more confrontational.
The Vatican also took a dim view of Trump's anti-Muslim campaign rhetoric, although he softened his tone considerably in a major speech in Riyadh.
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 8:30 a.m. 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.