We develop a new formulation of the quantum mechanics of a relativistic fermion based on the existence of an extended set of underlying classical coordinates. We enlarge the classical theory to a superspace built out of ordinary space-time coordinates and a set of Grassmann coordinates which transform like Majorana spinors and describe a classical spin. We introduce a natural supersymmetry between the Grassmann and the space-time coordinates. We quantize the theory both canonically and via a relativistic particle-mechanics path-integral prescription. The quantization of the motion of the coordinates in the superspace gives rise to the standard Dirac structure of a relativistic quantum-mechanical particle with spin one-half.