This Parish Prison complex was constructed in 1834 and closed in
about
1894.  It was bounded by Orleans, St. Ann, Treme and Marais
Streets
, near the location of present-day Louis Armstrong Park.  
This is the oldest photo I've found of the prison.
1864 Photo

Old Parish Prison
For more images of this prison and the one which came after it:

Old Orleans Parish Court and Old Parish Prison