We consider the distributed integer-weight-balancing problem in networks of nodes that are interconnected via directed edges, each able to admit a positive integer weight (or flow) within a certain interval, captured by lower and upper limits. We propose and analyze distributed iterative algorithms for obtaining admissible and balanced integer weights, i.e., integer weights within the given interval constraints, such that, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. The proposed algorithms assume that communication among pairs of nodes that are interconnected is bidirectional and are shown to lead to a set of admissible and balanced integer weights after a finite number of time steps (as long as the necessary and sufficient integer circulation conditions are satisfied on the given digraph).