Ariola et al defined a call-by-need λ-calculi with control, together with a sequent calculus presentation of it, and a mechanically generated continuation-passing-style transformation simulating the reduction. We present here a simply-typed version of this calculus and shows that it maps to System F through the continuation-passing-style transformation. This implies in particular the normaliza-tion of this simply-typed call-by-need calculus with control. Incidentally, we treat bound variables for the continuation-passing-style transformation in a precise way using indices rather than up to α-conversion, what makes it directly implementable.