Model-predictive control is an appealing framework to control robots due to its ability to exploit both sensory information and model predictions. But its performance remains fundamentally limited in tasks involving contact with the environment, in part because optimal control policies do not reason over force measurements. In this article, we propose a first complete answer to this issue by introducing a novel approach that systematically includes measured efforts into the optimal control loop. We propose to augment the state-space with a visco-elastic model of the contact force in the task space. We derive a complete predictive controller with an efficient formulation whose implementation is released in open-source. We conduct extensive comparisons with two other methods: the classical model-predictive control formulation, which inherently restricts the feedback to position and velocity information, and our previous approach that enabled torque feedback in the joint space. We demonstrate through simulation studies and hardware experiments, the benefit of exploiting Cartesian force measurements in the model-predictive control framework to achieve challenging contact tasks.