This paper shows how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit perfectly matched layers (PML) typically used as absorbing boundary conditions in computational electromagnetics codes. For the new PML equations, we give energy arguments that show the fields in the layer are bounded by a time-independent constant, hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields.