The electrical conductivity of biomaterials on a molecular scale is of fundamental interest in the life sciences. We perform first principles electronic structure calculations, which clearly indicate that lambda-DNA chains should present large resistance values. We also present two direct procedures to measure electrical currents through DNA molecules adsorbed on mica. The lower limit for the resistivity is 10(6) Omega . cm, in agreement with our calculations. We also show that low energy electron bombardment induces a rapid contamination and dramatically affects the measured conductivity, thus providing an explanation to recent reports of high DNA conductivity.