We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a flux function that depends discontinuously on the spatial position. We construct a weak solution to the sedimentation model by proving the convergence of a front tracking method. The basic building block in this method is the solution of the Riemann problem, which is complicated by the fact that the flux function is discontinuous. A feature of the convergence analysis is the difficulty of bounding the total variation of the conserved variable. To overcome this obstacle, we rely on a certain non-linear Temple functional under which the total variation can be bounded. The total variation bound on the transformed variable also implies that the front tracking construction is well defined. Finally, via some numerical examples, we demonstrate that the front tracking method can be used as a highly efficient and accurate simulation tool for continuous sedimentation.