The complex dielectric permittivity of sea ice controls the reflection, scattering, absorption and transmission of incident EM radiation, and is not only used for remote sensing and ice characterization, but underlies the optical properties that impact upon sea-ice heat and mass balance. At GHz frequencies the permittivity of sea ice is a focus of attention because of its importance for remote sensing, while at MHz and lower frequencies, for which the wavelength is significantly greater than the scale of features within the ice, it is related to both the geometry of brine inclusions and the brine volume fraction. As with the other physical properties of sea ice, in situ measurements on natural ice are difficult to obtain as most methods disturb the natural state. We have adapted the technique of d.c. cross-borehole tomography, recently used to make in situ measurements that characterize the anisotropic electrical resistivity of sea ice, to make a.c. measurements of the complex electrical conductivity of sea ice at a range of frequencies. Standard electromagnetic theory allows the complex conductivity to be related to the permittivity, hence allowing the variation of permittivity with frequency to be determined. We present results from both laboratory measurements made on artificially grown sea ice and trial field measurements made in McMurdo Sound.