This article proves the well posedness of the boundary value problem that arises when Bérenger's PML agorithm is applied to Pauli's equations. As in standard practice, the computational domain is rectangular and the absorptions are positive near the boundary and zero in the interior so are always x-dependent. At the boundary of the rectangle , the natural absorbing boundary conditions are imposed. The estimates allow exponential growth in time, but have no loss of derivatives for the physical quantities. The analysis proceeds by Laplace transform. Existence is proved for a carefully constructed boundary value problems for a complex stretched Helmholtz equation. Uniqueness is reduced by an analyticity argument to a result in [15]. This is the first stability proof for Bérenger's algorithm with x-dependent absorptions on a domain whose boundary is not smooth.