We study the existence of discrete breathers (time-periodic and spatially localized oscillations) in a chain of coupled nonlinear oscillators modelling the breathing of DNA. We consider a modification of the Peyrard-Bishop model introduced by Peyrard et al. [Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys. 35 (2009), 73-89], in which the reclosing of base pairs is hindered by an energy barrier. Using a new kind of continuation from infinity, we prove for weak couplings the existence of large amplitude and low frequency breathers oscillating around a localized equilibrium, for breather frequencies lying outside resonance zones. These results are completed by numerical continuation. For resonant frequencies (with one multiple belonging to the phonon band) we numerically obtain discrete breathers superposed on a small oscillatory tail.