Sen. John Boozman, R-Ark., said Trump's policy moves the US backward.
But, facing pressure from USA business and some of his fellow Republicans to avoid turning back the clock completely in relations with communist-ruled Cuba, he also will leave intact many of Obama's steps toward normalization. And, of course, nothing prevents the Cuban government from simply moving revenue over to the military or state security, a vulnerability in the policy that the White House has not addressed.
In his speech, Trump was quick to single out Cuba's human rights record under the "brutal" Castro regime of "oppressive" communism.
But "personally, part of what makes it hard [to accept] is that we were six years into the administration and spent a year and a half of exhaustive negotiations before announcing" the Cuba opening, said Rhodes, who coincidentally spoke at a Cuban entrepreneurship event in Miami on Monday. "Officially, today, they are rejected".
In practice, however, many recent changes to boost ties to Cuba will stay as they are.
However, despite the rhetoric, the order appears to be less far-reaching than the President claimed, for example, the embassies that opened in Havana and Washington will be maintained, Cuban Americans will be allowed to send money to their families and visit them, and US companies will be allowed to continue commercial transportation, including flights between the two countries.
"If you want Cuba to change and reform, we are doing the opposite of what would be most likely to bring about reforms", said Ben Rhodes, a former Obama aide who helped negotiate rapprochement.
"The last two years have shown that the two countries can cooperate and co-exist in a civilised way", it said.
But the directive leaves open embassies in Washington and Havana, and cruises and direct flights between the United States and Cuba will be protected under an exception from the prohibition on transactions with military-controlled entities. The Cuban News Agency said Trump took "a step backward" in ties with Cuba, by adopting a "unilateral" and "interventionist" stance.
The current Cuban sanctions regulations permit individuals to independently visit Cuba under the people-to-people travel authorization within the educational travel general license, without traveling as part of a group. The move shuts down what amounted to a backdoor way to allow American tourism in Cuba, despite the decades-old embargo that prohibits it.
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The changes won't go into effect until new documents laying out details are issued.
Members of Cuba's small but vibrant independent civil society say they fear the new policy will do more harm than good.
The officials spoke on condition of anonymity to discuss the policy before Trump announces it, despite the president's regular criticism of the use of anonymous sources.
Although diplomatic relations, restored only two years ago, will remain intact, Trump tightened rules for Americans travelling to Cuba, banned ties with a military-run tourism firm and reaffirmed the existing U.S. trade embargo. He said Obama's policies had helped Cubans.
Cuba has repeatedly said it will not renege on the promise of the former president, who died in November.
Yet it also exposed the shortcomings in Obama's approach.
Trump had framed his revision as a move against a "cruel and brutal" regime: bypassing the state military-run business group GAESA to channel investment to the people. But now we're being blocked again. Cuban Americans will still be able visit and send remittances to their families. Emmer, Crawford and five other House Republicans have warned that rolling back U.S. Cuba policy could threaten new bilateral agreements with Havana to combat human trafficking, illicit drugs and cyber crimes. And to understand these changes, we're joined by NPR's Carrie Kahn, who joins us from Havana.
In a memo from the Trump administration, it notes "any further improvements in the United States-Cuba relationship will depend entirely on the Cuban government's willingness to improve the lives of the Cuban people", and also mandates regular reporting of progress toward this objective.
Trump's new policy will directly limit commerce with GAESA, the Cuban military's business and commerce wing.
In December 2014, in the most sweeping change in U.S. "There's so much economic potential for Colorado industries such as agriculture, renewable energy and biotechnology that stand to benefit from the normalized relations with Cuba and now those opportunities are in doubt".
Aug 2016: U.S. commercial flight arrives in Cuba for the first time in more than half a century.
"America is prepared to outstretch its hand and work with the people of Cuba, but we will not, we will not empower their oppressors", Sen.
"The government of Cuba denounces the new measures toughening the embargo" imposed since 1962, according to a statement read on Cuban state television.
Information for this article was contributed by Michael Weissenstein, Andrea Rodriguez and Catherine Lucey of The Associated Press and by John Wagner and Michael Rosenwald of The Washington Post.