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Time domain integral equation (TDIE) solution methods have received more limited attention than other methods in computational electromagnetics because their early implementations were inaccurate, inefficient, and, worst of all, unstable. In spite of the advances made towards improving the efficiency of TDIEs, no consensus has so far been reached on how to stabilize TDIE solvers. The paper offers a novel and robust solution to the low frequency instability encountered in the implementation of currently available TDIE solvers. Specifically, the approach implements a loop-tree decomposition to the space of spatial testing functions, and treats the equations tested with solenoidal testing functions differently than the other equations. The underlying TDIE solver uses higher order divergence-conforming basis functions and bandlimited interpolatory functions (BLIFs) to effect, respectively, the spatial and temporal discretizations of the integral equation, and implements a bandlimited extrapolation technique to recover causality from the noncausal system generated by the BLIFs. Numerical results show that the proposed method is stable and exhibits exponential convergence with respect to a parameter of the temporal discretization.