Final answer:
The Franco-Mexican War coincided with the American Civil War, which allowed France to attempt to establish a monarchy in Mexico without initial U.S. intervention.
Step-by-step explanation:
The American war that coincided with the Franco-Mexican War was the American Civil War. The civil war lasted from 1861 to 1865 and overlapped with the Franco-Mexican War, which occurred from 1861 to 1867. During the Franco-Mexican War, the French, under Napoleon III, sought to establish a monarchy in Mexico, taking advantage of the fact that the United States was preoccupied with its own internal conflict and would be unlikely to enforce the Monroe Doctrine, which opposed European colonization in the Americas. However, once the Civil War ended, the U.S. began to exert pressure on France to withdraw its troops from Mexico, contributing to the eventual collapse of the French-backed Mexican Empire under Maximilian I.