Thermal contact resistance (TCR) is most commonly measured using one-dimensional steady-state calorimetric techniques. In these experimental methods, a temperature gradient is applied across two contacting beams and the temperature drop at the interface is inferred from the temperature profiles of the rods that are measured at discrete points. During data analysis, thermal conductivity of the beams is typically taken to be an average value over the temperature range imposed during the experiment. Here, a generalized theory is presented that accounts for temperature-dependent changes in thermal conductivity. The procedure presented enables accurate measurement of TCR for contacting materials whose thermal conductivity is any arbitrary function of temperature. For example, it is shown that the standard technique yields TCR values that are about 15% below the actual value for two specific examples of copper and silicon contacts. On the other hand, the generalized technique predicts TCR values that are within 1% of the actual value. The method is exact when thermal conductivity is known exactly and no other errors are introduced to the system.