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Apparatus for measurement of the temperature dependence of the thermal conductivity of small specimens in the range of 8-250 K is described. The method consists of reproduction of temperature fields in an experimental cell with and without a heat-transfer channel through the specimen by thermal conductivity. This makes it possible to measure heat losses through the structural elements and losses due to radiation from the specimen surface. The method is illustrated by measurement of the thermal conductivity of a filamentary crystal with dimensions of 5*28*1300 mu m. The random error is 20% for T< 10 K and is reduced to 3% for T> 100 K. The low specific heat of the sensing element, which is 22 mu J/K at T=250 K, allows the apparatus to be used as a microcalorimeter.