DNA wrapping is an important mechanism for chromosomal DNA packaging in cells and viruses. Previous studies of DNA wrapping have been performed mostly on torsionally unconstrained DNA, while in vivo DNA is often under torsional constraint. In this study, we extend a previously proposed theoretical model for wrapping of torsionally unconstrained DNA to a new model including the contribution of DNA twist energy, which influences DNA wrapping drastically. In particular, due to accumulation of twist energy during DNA wrapping, it predicts a finite amount of DNA that can be wrapped on a helical spool. The predictions of the new model are tested by single-molecule study of DNA wrapping under torsional constraint using magnetic tweezers. The theoretical predictions and the experimental results are consistent with each other and their implications are discussed.