We present a high-resolution microwave spectrometer to measure the frequency- dependent complex conductivity of a superconducting thin film near the critical temperature. The instrument is based on a broadband measurement of the complex reflection coefficient, S11, of a coaxial transmission line, which is terminated to a thin film sample with the electrodes in a Corbino disk shape. In the vicinity of the critical temperature, the standard calibration technique using three known standards fails to extract the strong frequency dependence of the complex conductivity induced by the superconducting fluctuations. This is because a small unexpected difference between the phase parts of S11 for a short and load standards gives rise to a large error in the detailed frequency dependence of the complex conductivity near the superconducting transition. We demonstrate that a new calibration procedure using the normal-state conductivity of a sample as a load standard resolves this difficulty. The high quality performance of this spectrometer, which covers the frequency range between 0.1 and 10GHz, the temperature range down to 10K, and the magnetic field range up to 1T, is illustrated by the experimental results on several thin films of both conventional and high temperature superconductors.