We present a new approach for solving the Signorini problem with Coulomb friction in linear elasticity. The Signorini problem describes the linearized contact of an elastic body with a rigid foundation. Because it is not known in advance which part of the body's surface will be in contact with the foundation, the corresponding energy functional is piecewise smooth but not differentiable on the contact boundary. In order to solve the Signorini problem with Coulomb friction, a sequence of Signorini problems without friction has to be solved. Thus, we focus first on solving the frictionless problem.