“The world has changed, and the world has changed particularly in Latin America,” said a senior official, on the condition of anonymity in order to speak candidly about diplomatic issues. The official was alluding to an absence of the kind of insurgencies that Cuba once supported, activity that led to its placement on the list in 1982.

Cuban officials have said they would find it hard to move forward with diplomatic relations while remaining on the list, which they see as a blemish to their nation’s image and a scarlet letter that has blocked Cuba from doing business with American banks and led some international institutions to shy away from opportunities to work with Cuba.

Not even Cuba’s interests section in Washington, the outpost that performs some functions of an embassy, could get a bank account as financial institutions worried about violating sanctions from the Treasury Department over doing business with a state on the terrorism list and running afoul of the trade embargo.

In speaking to reporters, another senior administration official, however, said it appeared that Cuba had found a bank even before the announcement, in part because of Treasury Department “steps to ease the situation and facilitate” an agreement.

The United States had sought to keep the terrorism designation question separate from the issue of restoring diplomatic relations, focusing its demands on ensuring that its diplomats could travel freely in Cuba and that Cubans would not be bothered by the police as they entered the redesignated American Embassy.

In a sign of the diplomatic thaw, Cuba attended the summit meeting for the first time since the gathering’s inception in 1994.

The meeting created the first publicly planned encounter of the American and Cuban presidents since 1958, though Mr. Obama and Mr. Castro shook hands at Nelson Mandela’s memorial service in South Africa in December 2013 and President Bill Clinton and Fidel Castro shook hands and chatted briefly at a United Nations meeting in 2000.