If by "end" you mean returned unemployment to exactly pre-1929 levels then, on the basis of that remarkably limtied definition, you are correct. Unemployment had, however, declined dramatically from the month FDR took office until Dec. 1941, at which time unemployment was not far above pre-depression levels. In fact unemployment probably would've reached that level had FDR not ignored his Keynesian advisers and impose austerity measures which directly led to the 1937-39 recession.
Some of that might have been the "Arsenal of Democracy" effect. 12 months before, America was selling to the increased demand.
Keynesian policies may a diminishing marginal utility effect when it comes to increasing demand (and yes, this thread might be more appropriate in the economics section).
Unemployment plummetted dramatically upon institution of New Deal policies until FDR temprarily abandoned Keynesianism in 1937. More importantly, armament production during America's period of non-involvement in \WWIIO was litterally a drop in the bucket compared to post Pearl Harbor wartie spending.
But then, isn't the point that unemployment dropped even further--to near full employment--by virtue of
government spending policies the point? The fact that it was done in that instance \through defense spending and in the early NewDeal days through social programs and public works says less about the ability of one versus the other's ability to alleviate unemployment, and more about conservatives' view of the "proper" limits and role of government sending.