A one-dimensional two-phase Stefan problem is examined for the melting of a semi-infinite material with thermal conductivity that depends on temperature following a power-law relationship. The choice to model thermal parameters as functions of temperature is rooted in both physical and industrial considerations, providing a more accurate and realistic representation of phase change processes. By applying a Dirichlet type boundary condition at the fixed face, a theoretical analysis is conducted. A similarity transformation is used to convert the problem into an equivalent ordinary differential problem, from which an integral problem is derived. The existence of at least one analytic solution is assured through the application of the Banach fixed-point theorem.