Thermoelectric oxide nanofibers prepared by electrospinning are expected to have reduced thermal conductivity when compared to bulk samples. Measurements of nanofibers' thermal conductivity is challenging since it involves sophisticated sample preparation methods. In this work, we present a novel method suitable for measurements of thermal conductivity of a single nanofiber. A microelectro-mechanical (MEMS) device has been designed and fabricated to perform thermal conductivity measurements on a single nanofiber. A special Si template was designed to collect and transfer individual nanofibers onto a MEMS device. Pt was deposited by a focused ion beam to reduce the effective length of a prepared nanofiber. La0.95Sr0.05CoO3 nanofibers with diameters of 140 nm and 290 nm were studied and characterized using this approach at room temperature. Measured thermal conductivities yielded values of 0.7 W center dot m(-1)center dot K-1 and 2.1 W center dot m(-1)center dot K-1, respectively. Our measurements in La0.95Sr0.05CoO3 nanofibers confirmed that a decrease of linear dimensions has a profound effect on its thermal conductivity.