PANAMA CITY — President Barack Obama met Cuban President Raul Castro on Saturday in the highest-level talks between the two countries in nearly 60 years, and the two men agreed to push ahead on improving relations after decades of hostility.

Describing their private meeting as “historic,” Obama said the two countries can now end the antagonism of the Cold War era, although he said he would continue to pressure the communist-led country on democracy and human rights.

“Obviously, there are still going to be deep and significant differences between our two governments,” Obama told Castro as they met in Panama, where they both attended a summit of leaders from across the Americas.

Obama said he believed both sides could raise their concerns about the other’s policies yet still work together to boost commercial, travel and diplomatic ties.

“The Cold War is over … Cuba is not a threat to the United States,” Obama later told reporters, pointing out that at 53, he wasn’t even born when Castro and his brother Fidel seized power in the 1959 Cuban revolution.

The meeting followed a landmark agreement in December, when Obama and Castro announced they would move to normalize relations, including seeking to restore diplomatic ties that were broken off by Washington in 1961.

Obama said he decided to overturn longstanding U.S. policy on Cuba because the old approach of open hostility and economic sanctions had failed to force through major changes on the Caribbean island and it was time to try something new.

Since then, he has relaxed some restrictions on travel and trade with Cuba, although a longstanding economic embargo remains in place.

At their 80-minute meeting on Saturday, almost unimaginable until recently, Obama and Castro sat side by side in polished, wooden chairs in a small conference room. The mood was described by Obama aides as cordial but businesslike.

Both men wore dark suits and each nodded and smiled as the other spoke. Obama towered over Castro when they stood up to shake hands in front of reporters before starting their meeting.

Castro said he would continue to take steps toward normalizing relations, and was open to discussing human rights and other issues.

“So we are willing to discuss everything, but we need to be patient, very patient. Some things we will agree on; others we will disagree,” said the 83-year-old leader, who took over as president of Cuba in 2008 when Fidel Castro stepped aside because of ill health.

During a summit session earlier, Castro apologized to Obama for a series of impassioned broadsides against the U.S. for its Cold War attempts to topple communist rule on the island.

He said the U.S. leader was not to blame for any of those policies of the past, and called him “an honest man.”

Castro has already undertaken some market-style reforms to try to strengthen Cuba’s economy but he is moving cautiously and he has made clear that he has no intention of allowing an end to communist rule.

Obama has faced some criticism inside Congress for his dramatic shift on Cuba policy. Critics say he has given up too much without first insisting on political reform on the island.

The last time the leaders of the two countries held a substantive meeting was in 1956, when Dwight Eisenhower was president and Fulgencio Batista was the U.S.-backed dictator in power in Havana.

The Castro brothers toppled Batista in a revolution on January 1, 1959, and relations between the U.S. and Cuba quickly deteriorated.

Fidel Castro became a Cold War ally of the Soviet Union and the rivalries took the world to the brink of a nuclear war during the Cuban Missile Crisis in October, 1962.

As relations now improve, Obama is close to removing Cuba from a U.S. list of countries that it says sponsor terrorism. Raul Castro’s government has called Cuba’s inclusion on the list a hindrance to restoring diplomatic ties.

A senior U.S. official said Obama will make his decision in coming days. Lawmakers are extremely unlikely to block the move.

Since 1977, the two countries have maintained contact through interests sections in Havana and Washington and in recent years they have cooperated on migration, drug trafficking and other issues.

Cuba was placed on the U.S. terrorism list in 1982 when it was supporting Marxist rebel groups in Latin America but that backing stopped with the end of the Cold War.

Obama said on Saturday that his new Cuba policy has majority support among Americans and the overwhelming backing of Cubans.

In the U.S., polls show that support for a more collaborative relationship with Havana has grown steadily in recent years, even in the Cuban-American community that for decades fiercely opposed Fidel Castro and overwhelmingly opposed ties with Havana.

Obama can continue to ease specific sanctions. But the overall trade embargo against Cuba can be overturned only by the Republican-controlled Congress.

The president’s changes to Cuba policy are part of a broader approach of being open to finding common ground with traditional foes of the U.S. Despite criticism from opponents, his government has pushed for the negotiations between Iran and major powers aimed at curbing Tehran’s nuclear program.

Speaking at the summit earlier on Saturday, Obama challenged Latin American leaders to improve human rights and democracy and stop using past U.S. policies as “a handy excuse” for political problems they face at home, a clear reference to leftist governments in Venezuela, Cuba, Ecuador and others.

“The Cold War has been over for a long time. I’m not interested in having battles that frankly started before I was born,” Obama said. “What I am interested in is solving problems and working with you.”