One battle, not the war. The British won most of the other engagements. The American victory at New Orleans did not substantially affect the outcome of the war because it occurred after the treaty was signed and the loss by the British did not disturb the status quo ante anticipated by the treaty. An American loss, on the other hand, could have had some repercussions because, under the terms of the treaty, Britain would be within its rights to hold New Orleans because it would have been conquered before Congress ratified the treaty. Whether the British would have tried to keep New Orleans is a question for counter-factual history.
