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A new perfectly matched layer (PML) formulation for the time-domain finite element method is described and tested for problems in three-dimensional space. An integral part of this PML algorithm is a novel time integration scheme based on Galerkin's method with a piecewise linear temporal expansion of the electric field. Our time integration procedure is constructed by forming a linear combination of exact and trapezoidal integration applied to the temporal weak form, which reduces to the well-known Newmark scheme in the case where the PML is not present. This technique allows for a consistent time discretization of the electric field vector wave equation augmented with the PML, which includes terms that involve the electric field itself, time derivatives thereof and exponential functions temporally convolved with the electric field. The auxiliary variables that are associated with such convolutions are efficiently updated in an explicit manner by matrix-vector products. The new PML formulation is successfully tested on canonical problems. The PML can very efficiently absorb propagating electromagnetic waves, which is demonstrated by reflections coefficients as low as -100 dB. The PML formulation is capable of giving highly accurate results for very broad frequency bands which is confirmed by tests that show an error smaller than 2% over more than six octaves in frequency. Finally, we apply the PML formulation to a bow-tie antenna and comparisons with measurements show good agreement.