A new method of high-precision isotherm thermometry has been developed in which the density of helium gas is determined from dielectric constant measurements using the virial expansion of the Clausius-Mossotti equation. The technique is shown to be accurate at the level of similar 0.3 mK for the range 4.2-27.1 K, with potential for both increased accuracy and increased range. The results confirm that the gas thermometer scale NPL-75 is internally consistent to better than 1 mK, and that Berry's recent values for the second virial coefficient of helium are correct at the level similar ± 0.05 cm³ mol^-1. The results also give a value for the third virial coefficient, C(T) = (5420 ± 225)/T cm⁶ mol^-2, and a value of 517.257 ± 0.025 × 10^-3 cm³ mol^-1 for the polarizability of ⁴He.