The paper is devoted to the numerical solution of the ill-posed Cauchy problem for the Laplace equation by means of the conjugate gradient method implemented with the aid of the boundary-element method. This problem formulates as follows: \[ \Delta u= 0,\quad x\in\Omega,\quad u|_{\Gamma_1}= \phi,\quad {\partial u\over\partial n}\biggl|_{\Gamma_1}= g, \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^n\) and \(\Omega\) consists of two non-intersecting \((n-1)\)-dimensional manifolds \(\Gamma_1\) and \(\Gamma_2\). The above problem is transformed in a variational one whose solution is approximated with a conjugate gradient method with a stopping rule. Numerical experiments carried out with a BEM method show the convergence of the approximation method.