We present a Faraday rotation measure (RM) study of the diffuse, polarized, radio emission from the giant lobes of the nearest radio galaxy, Centaurus A. After removal of the smooth Galactic foreground RM component, using an ensemble of background source RMs located outside the giant lobes, we are left with a residual RM signal associated with the giant lobes. We find the most likely origin of this residual RM is from thermal material mixed throughout the relativistic lobe plasma. The alternative possibility of a thin-skin/boundary layer of magnetoionic material swept up by the expansion of the lobes is highly unlikely since it requires, at least, an order of magnitude enhancement of the swept up gas over the expected intragroup density on these scales. Strong depolarisation observed from 2.3 to 0.96 GHz also supports the presence of a significant amount of thermal gas within the lobes; although depolarisation solely due to RM fluctuations in a foreground Faraday screen on scales smaller than the beam cannot be ruled out. Considering the internal Faraday rotation scenario, we find a thermal gas number density of ~10^{-4} cm^{-3} implying a total gas mass of ~10^{10} M_solar within the lobes. The thermal pressure associated with this gas (with temperature kT ~ 0.5 keV, obtained from recent X-ray results) is approximately equal to the non-thermal pressure, indicating that over the volume of the lobes, there is approximate equipartition between the thermal gas, radio-emitting electrons and magnetic field (and potentially any relativistic protons present). ; Comment: Accepted for publication in The Astrophysical Journal. 10 pages, 7 figures, 2 tables