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This paper presents a methodology for protecting low-priority best-effort (BE) traffic in a network domain that provides both virtual-circuit routing with bandwidth reservation for QoS traffic and datagram routing for BE traffic. When a QoS virtual circuit is established, bandwidths amounting to the traffic's effective bandwidths are reserved along the links. We formulate a new QoS-virtual-circuit admission control and routing policy that sustains a minimum level of BE performance. In response to a QoS connection request, the policy executes a two-stage optimization. The first stage seeks a minimum-net-effective-bandwidth reservation path that satisfies a BE protecting constraint; the second stage is a tie-breaking rule, selecting from tied paths one that least disturbs BE traffic. Our novel policy implementation efficiently executes both optimization stages simultaneously by a single run of Dijkstra's algorithm. According to simulation results, within a practical operating range, the consideration that our proposed policy gives to the BE service does not increase the blocking probability of a QoS connection request.