In this paper, we present the FB (Fair Buffering) mechanism for the efficient support of TCP traffic over UBR connections in the subnet ATM model. We show that both throughput and fairness may be improved by using this congestion control mechanism. The FB mechanism is founded on the observation that the performance of TCP over UBR connections is optimal when the buffer space is allocated in proportion to the connection's bandwidth-delay product. We compare the performance of the existing drop-tail, EB (Equal Buffering) and the proposed FB buffer management schemes, with and without RR (Round-Robin) scheduling, and show through simulation the effectiveness of the proposed FB mechanism when used with the RR scheduling scheme.