For the measurement of the thermal conductivity of the formations surrounding a borehole, a sonde having variable diameter (consisting of an inflatable rubber cylinder with heating wires embedded in its wall) is described. The conditions for the usual sonde made of metal are no longer fulfilled, but the solution to the problem of determining the thermal conductivity from the temperature rise is given, based on an approach by Carslaw and Jaeger, which contains the Bessel functions of the second kind. It is shown that a simpler solution for large values of time can be obtained through the Laplace transformation, and the necessary series developments for computer application are also given. The sonde and the necessary measuring circuitry are described. Tests measurements have indicated that the thermal conductivity can be determined with this sonde with a precision of + 10%.