An optimal control problem for the 1-d heat equation is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear functions. The connection between the solutions of the discretized problems and the continuous one is investigated. Under an additional assumption on the adjoint state an approximation order σ^3/2 is proved for uniform discretizations. In the general case it is shown that a non-uniform control discretization ensures an approximation of order σ^3/2. Numerical tests confirm the theoretical part.