We introduce and study a diffuse interface model describing cell motility. We provide a detailed rigorous analysis of the model in dimension 1 and formally derive the sharp interface limit in any dimension. The model integrates the most important physical processes involved in cell motility, such as incompressibility, internal stresses exerted by the cytoskeleton seen as an active gel, and dynamic contact lines. The resulting nonlinear system couples a degenerate fourth order parabolic equation of Cahn-Hilliard type for the phase variable with a convection-reaction-diffusion equation for the active potential. The sharp interface limit leads to a Hele-Shaw type free boundary problem which includes the effects of surface tension and an additional destabilizing term at the free boundary. This additional term can be seen as a nonlinear Robin type boundary condition with the "wrong" sign. Such a boundary condition reflects the active nature of the cell, e.g., protrusion formation. We rigorously investigate the properties of this model in one dimension and prove the appearance of nontrivial traveling wave solutions for the limit problem when the key physical parameter exceeds a certain critical value. Although minimal, this new Hele-Shaw model, with Robin's unconventional boundary condition, is rich enough to describe the universal property of migrating cells that has been recently described by various theoretical biophysical models.