Cuba's communist economy is sinking, but its capitalist entrepreneurs are rising — and the U.S. wants to partner with them before Russia does. Critics, meanwhile, call the Cuban private sector a hoax.

HAVANA — The house on Villegas Street, in the heart of Old Havana, looks nothing like the stately two-story home it used to be a century ago, with its high ceilings, wrought iron railings, ...