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The War of 1812 is the correct answer as it is the only war listed that was fought during the 1800s. This conflict occurred between 1812 and 1815, primarily involving the United States and Great Britain. It arose out of issues such as trade restrictions and impressment of American sailors by the British navy.

In contrast, World War I took place from 1914 to 1918 and World War II spanned from 1939 to 1945, both of which fall well outside the 19th century. Operation Desert Storm, part of the Gulf War, occurred in 1991, also outside the specified timeline. Therefore, the War of 1812 is the only choice that accurately identifies a war from the 1800s.