At frequencies of about 10 GHz, the complex relative permittivity of liquids of low loss is determined by an automatic precision method based on the computer-controlled measurement of the microwave power at the lower exit of a liquid-filled, inclined, standard waveguide and on a computer curve-fitting procedure. For the real part of the permittivity, a relative uncertainty of ±0.0007 is achieved. For the loss tangent, the uncertainty is |δtan Δ| ≦ 11 × 10-6.