We consider an inverse heat conduction problem (IHCP) in a quarter plane. We want to know the distribution of surface temperature in a body from a measured temperature history at a fixed location inside the body. This is a severely ill-posed problem in the sense that the solution (if exists) does not depend continuously on the data. Eldén (1995) has used a difference method for solving this problem, but he did not obtain the convergence at x = 0 . In this paper, we gave a logarithmic stability of the approximation solution at x = 0 under a stronger a priori assumption ‖ u ( 0 , t ) ‖ p ≤ E with {1}/{2}$"> p > 1 / 2 . A numerical example shows that the computational effect of this method is satisfactory.