High thermal conductivity materials are useful for thermal management applications and fundamental studies of phonon transport. Past measurements of several ultrahigh thermal conductivity materials were not able to obtain the peak thermal conductivity, which is expected to appear at a low temperature and contains insight into the competition between extrinsic phonon-defect and phonon-boundary scattering with intrinsic phonon-phonon processes. Here, we report a peak thermal conductivity measurement method based on differential Wheatstone bridge measurement of the small temperature drop between two thin film resistance thermometers patterned directly on the sample. With the use of a mesoscale silicon bar sample as the calibration standard, this method is able to obtain results that agree with past measurements of large bulk silicon crystals at high temperatures and first-principles calculation results that account for additional phonon-boundary scattering in the sample. The agreement demonstrates the accuracy of this measurement method for peak thermal conductivity measurements of high thermal conductivity materials.