In this paper, the method of lines approximation for a rather general elliptic equation containing a diffusion coefficient is considered. Our main results are the regularization of the ill-posed Cauchy problem and the proof of error estimates leading to convergence results for the method of lines. These results are based on the conditional stability of the continuous Cauchy problem and the approximation by appropriately chosen finite-dimensional spaces, onto which the possibly perturbed Cauchy data are projected. At the end of this paper, we present and discuss results of some of our numerical computations. There are multiple applications in material sciences, thermodynamics, medicine etc.; related problems are shape optimization problems which are important, e.g. for nondestructive testing, crack location, thermal tomography, and other applications.