Collisions of 0-4 eV electrons with thin DNA films are shown to produce single strand breaks. The yield is sharply structured as a function of electron energy and indicates the involvement of pi* shape resonances in the bond breaking process. The cross sections are comparable in magnitude to those observed in other compounds in the gas phase in which pi* electrons are transferred through the molecule to break a remote bond. The results therefore support aspects of a theoretical study by Barrios et al. [J. Phys. B 106, 7991 (2002)]] indicating that such a mechanism could produce strand breaks in DNA.