It is known that a data network may not be stable at the connection level under some unfair bandwidth allocation policies, even when the normal offered load condition is satisfied, i.e., the average traffic load at each link is less than its capacity. In this note, we show that, under the normal offered load condition, a data network is stable when the bandwidth of the network is allocated so as to maximize a class of general utility functions. Using the microscopic model proposed by Kelly for a transmission control protocol (TCP) congestion control algorithm, we argue that the bandwidth allocation in the network dominated by this algorithm can be modeled as our bandwidth allocation model, and hence that the network is stable under the normal offered load condition. This result may shed light on the stability issue of the Internet since the majority of its data traffic is dominated by the TCP.