A process for hydroprocessing a distillate stream to produce a stream exceptionally low in sulfur, with total aromatics and polynuclear aromatics being moderately reduced. A distillate stream is hydrodesulfurized in a first hydrodesulfurization stage. The product stream thereof is passed to a first separation stage wherein a vapor phase product stream and a liquid product stream are produced. The liquid phase product stream is passed to a second hydrodesulfurization stage and the product stream thereof is passed to a second separation stage wherein a vapor phase product stream and a liquid product stream low in sulfur are produced. At least a portion of the vapor product stream from said second separation stage can be cascaded to the first hydrodesulfurization stage.