The electrodynamic response of DNA in the millimeter wave range is investigated. By performing measurements under a wide range of humidity conditions and comparing the response of single-stranded DNA and double-stranded DNA, we show that the appreciable ac conductivity of DNA is not due to photon-assisted hopping between localized states but is instead due to dissipation from dipole motion in the surrounding water helix. Such a result, where the conductivity is due to the constrained motion of overdamped dipoles, reconciles the vanishing dc conductivity of DNA with the considerable ac response.