The fair distribution of bandwidth to different connections is an important issue in high-speed networks. This is especially true in low-priority services where the bandwidth available for the low priority connections may be small and may vary rapidly. An example is the Available Bit Rate (ABR) service in Asynchronous Transfer Mode (ATM) networks. This study uses the ABR service to explore a scheme that achieves better fairness in bandwidth allocations. Traditional schemes fail to achieve desired bandwidth allocations, which are generally based on connection weights. In this paper, a new scheme called Active Fairness is proposed which substantially improves fairness in bandwidth allocations. Contrary to one set of weights used in traditional schemes, Active Fairness maintains two sets of weights at each link. The first set is the same as that used in traditional schemes, which represents the relative bandwidth allocation desired out of the available bandwidth at the link. If these weights are used directly in bandwidth allocation, the actual proportion of the bandwidth utilized by an individual connection can deviate substantially due to factors such as the number of links traversed by the connection, its round-trip propagation delay, etc. In order to take into account these factors, another set of weights is derived at each link, which is used in bandwidth allocation. Simulation results under several different network conditions demonstrate the advantages of Active Fairness.