A half-century after Washington severed relations with Cuba, the United States’ seven-story mission looms over Havana’s seaside Malecon boulevard as the largest diplomatic outpost in the country. Cuban guards stand at close intervals on the street outside, and islanders line up by the thousands each year for a shot at a coveted visa.

The gleaming U.S. Interests Section suddenly is poised to become an even more important presence in Cuba as the two countries negotiate the first phase of their historic detente—transforming the complex into a full embassy that would reflect the Obama administration’s hopes of new influence on the communist island.