President Obama’s surprise effort to restore diplomatic relations with Cuba, concurrent with an economic embargo, recalls the two-track approach — economic and sometimes military force, along with secret, sporadic attempts to find some kind of accommodation — that formed American policy toward Cuba during the most dangerous years of that relationship.

On Monday evening, Nov. 18, 1963, at the Americana Hotel in Miami Beach — four days before his assassination — President John F. Kennedy, wearing black tie, told the Inter-American Press Association that only one issue separated the United States from Fidel Castro’s Cuba: Castro’s “conspirators” had handed Cuban sovereignty to “forces beyond the hemisphere” (meaning the Soviet Union), which were using Cuba “to subvert the other American republics.” Kennedy said, “As long as this is true, nothing is possible. Without it, everything is possible.”

The president had asked his speechwriter, Theodore Sorensen, for language that would open a door to the Cuban leader, although, as Sorensen later observed, the audience was “a very tough anti-Castro group.”

That same day, Ambassador William Attwood, a Kennedy delegate to the United Nations, secretly called Castro’s aide and physician, Rene Vallejo, to discuss a possible secret meeting in Havana between Attwood and Castro that might improve the Cuban-American relationship, which had been ruptured when President Eisenhower broke diplomatic ties in January 1961.

Attwood had been told by Castro’s U.N. ambassador, Carlos Lechuga, in September 1963, that the Cuban leader wished to establish back-channel communications with Washington. Attorney General Robert F. Kennedy worried that such talks would leak and embarrass his brother on the eve of his 1964 re-election campaign, but the president quietly encouraged Attwood to pursue the matter...