This paper introduces an original 1-fluid method for direct simulation of the motion of rigid particles in fluids. The model is based on the implicit treatment of a single fictitious fluid over a fixed grid, and uses an augmented Lagrangian optimization algorithm for the velocity–pressure coupling. The paper focuses on the case of a rigid sphere settling in a viscous medium. For validation purposes, simulations of the transient motion of a sedimenting sphere at Reynolds numbers ranging from 1.5 to 31.9 are compared to the PIV data published by Ten Cate et al. [Ten Cate A, Nieuwstad CH, Derksen JJ, Van den Akker HEA. Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity. Phys Fluids 2002;14(11):4012–25]. Accurate reproduction of the experimental data is obtained. Further simulations are intended to investigate higher Reynolds numbers. Predictions of transient particle sedimentation at Reynolds number 280 are performed and compared with experimental data of the sedimentation trajectory, as well as with simulation results based on the lattice-Boltzmann method.