Physical human-robot interaction (pHRI) demands simultaneous trajectory accuracy and compliant safety under unplanned contact. Classical impedance control incurs a nonzero steady-state position error under sustained human force -- the applied force divided by the task stiffness -- which integral action reduces only within a narrow stable-gain budget. We present a two-layer Impedance MPC that resolves this tension. Layer~1 analytically cancels gravity, Coriolis, and task-space inertia, reducing the residual plant to a configuration-independent double integrator with a constant state-transition matrix. Layer~2 solves a 30-variable convex QP at 100\,Hz, exploiting this constant structure so the free-response matrix is precomputed once; an augmented Kalman filter estimates the persistent disturbance state, giving a formal zero-steady-state-error guarantee. A null-space inverse-barrier potential and a task-space workspace projection enforce joint-limit safety across the tested workspace. On a 7-DOF Franka FR3, Impedance MPC with Kalman augmentation attains sub-0.05\,mm steady-state error versus 44.8\,mm for classical impedance (a $>$800-fold reduction) under a sustained 15\,N force, sub-millimeter tracking on four 3-D circles, and graceful robustness to measurement noise and inertial mismatch up to 30\%.