Thermal conductivity of one-dimensional nanostructures, such as nanowires, nanotubes, and polymer chains, is of significant interest for understanding nanoscale thermal transport phenomena as well as for practical applications in nanoelectronics, energy conversion, and thermal management. Various techniques have been developed during the past decade for measuring this fundamental quantity at the individual nanostructure level. However, the sensitivity of these techniques is generally limited to 1 × 10(-9) W∕K, which is inadequate for small diameter nanostructures that potentially possess thermal conductance ranging between 10(-11) and 10(-10) W∕K. In this paper, we demonstrate an experimental technique which is capable of measuring thermal conductance of ∼10(-11) W∕K. The improved sensitivity is achieved by using an on-chip Wheatstone bridge circuit that overcomes several instrumentation issues. It provides a more effective method of characterizing the thermal properties of smaller and less conductive one-dimensional nanostructures. The best sensitivity experimentally achieved experienced a noise equivalent temperature below 0.5 mK and a minimum conductance measurement of 1 × 10(-11) W∕K. Measuring the temperature fluctuation of both the four-point and bridge measurements over a 4 h time period shows a reduction in measured temperature fluctuation from 100 mK to 0.6 mK. Measurement of a 15 nm Ge nanowire and background conductance signal with no wire present demonstrates the increased sensitivity of the bridge method over the traditional four-point I-V measurement. This ultra-sensitive measurement platform allows for thermal measurements of materials at new size scales and will improve our understanding of thermal transport in nanoscale structures.