We consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain. A linear finite element method is used. It is shown that the standard method for solving the resulting linear equations in finite-precision arithmetic can give an arbitrarily inaccurate answer because of ill-conditioning in the stiffness matrix. A new method for solving the linear equations is proposed. This method is based on a "mixed formulation" and gives a numerically accurate answer independent of the variation in the coefficients. The numerical error in the solution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation.