The distance vector routing protocol satisfies the following property. If this protocol is used to route a sequence of data messages from a source to a destination, then these data messages will follow the same shortest-distance path from the source to the destination. In this paper, we show how to modify this protocol, without adding new messages, in order to satisfy the following load balancing property. If the modified protocol is used to route a sequence of data messages from a source to a destination, then these messages will be uniformly distributed over all k-monotonic paths from the source to the destination, i.e., over all paths whose distance to the destination never increases at each hop, and may remain constant in at most k hops. In particular, by choosing k to be zero, the modified protocol distributes the data messages over all shortest-distance paths from the source to the destination.