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We present a new method for measuring thermal conductivities of films with nanoscale thickness. The method combines a micro electrothermal test structure with a finite-element- based data analysis procedure. The test device consists of two serpentine nickel structures, which serve as resistive heaters and resistance temperature detectors, on top of the sample. The sample is supported by a silicon nitride membrane. Analytical solution of the heat flow is infeasible, making interpretation of the data difficult. To address this, we use a finite-element model of the test structure and apply nonlinear least-squares estimation to extract the desired material parameter values. The approach permits simultaneous extraction of multiple parameters. We demonstrate our technique by simultaneously obtaining the thermal conductivity of a 280 mum x 80 mum x 140 nm thick aluminum sample and the 360 mum x 160 mum x 180 nm thick silicon nitride support membrane. The thermal conductivity measured for the silicon nitride thin film is 2.1 W/mK, which is in agreement with reported values for films of this thickness. The thermal conductivity of the Al thin film is found to be 94 W/mK, which is significantly lower than reported bulk values and consistent both with reported trends for thin metallic films and with values that were obtained using electrical resistivity measurements and the Wiedemann-Franz law.