We consider a non-linear two-phase unidimensional Stefan problem, which consists on a solidification process, for a semi-infinite material x > 0, with phase change temperature T₁, an initial temperature T₂ > T₁ and a convective boundary condition imposed at the fixed face x = 0 characterized by a heat transfer coefficient h > 0. We assume that the volumetric heat capacity and the thermal conductivity are particular nonlinear functions of the temperature in both solid and liquid phases and they verify a Storm-type relation. A certain inequality on the coefficient h is established in order to get an instantaneous phase change process. We determine sufficient conditions on the parameters of the problem in order to prove the existence and uniqueness of a parametric explicit solution for the Stefan problem.