Summary: A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for \(y=0\), and boundary data are for \(x=0\) and \(x=1\). The elliptic operator is self-adjoint with nonconstant, smooth coefficients. The solution for \(y=1\) is sought. This Cauchy problem is ill-posed in an \(L^2\)-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle.