We consider networks the nodes of which are interconnected via directed edges, each able to admit a flow (or weight) within a certain interval, with nonnegative end points that correspond to lower and upper flow limits. This paper proposes and analyzes a distributed algorithm for obtaining admissible and balanced flows, i. e., flows that are within the given intervals at each edge and are balanced (the total in-flow equals the total out-flow) at each node. The algorithm can also be viewed as a distributed method for obtaining a set of weights that balance a weighted digraph for the case when there are lower and upper limit constraints on the edge weights. The proposed iterative algorithm assumes that communication among pairs of nodes that are interconnected is bidirectional (i. e., the communication topology is captured by the undirected graph that corresponds to the network digraph), and allows the nodes to asymptotically (with geometric rate) reach a set of balanced feasible flows, as long as the (necessary and sufficient) circulation conditions on the given digraph, with the given flow/weight interval constraints on each edge, are satisfied. We also provide a methodology that can be used by the nodes to asymptotically determine, in a distributedmanner, when the circulation conditions are not satisfied (thus, making the problem infeasible). Finally, we provide several examples and simulation studies to highlight the role of various parameters involved in the proposed distributed algorithm.