An algorithm for the solution of branching one-dimensional cable neuron models is presented. The algorithm is based on solving the finite-difference approximations to a cable or compartmental model of a neuron with a time implicit integration scheme. The algorithm solves the linear system of equations that must be solved at each time step with implicit algorithms via an "exact domain decomposition." This domain decomposition allows the solution of the unbranched and branching regions of the neuron to be done separately and permits a wide variety of possible implementations on parallel computers. Similarly, the separation of the straight and branched regions allows the solution of these two problems to be accomplished with linear system algorithms optimized for each class of problems. In contrast to other widely used methods (Hines, M. (1984) Int. J. Biomed. Comput., 15: 69-75), this algorithm can be used with arbitrary branching geometries, even those which contain closed loops.