Quantum key distribution1 has reached the level of maturity required for deployment in real-world scenarios2,3,4,5,6. It has previously been shown to operate alongside classical communication in the same telecommunication fibre7,8,9 and over long distances in fibre10,11 and in free-space links12,13,14,15. Despite these advances, the practical applicability of quantum key distribution is curtailed by the fact that most implementations and protocols are limited to two communicating parties. Quantum networks scale the advantages of quantum key distribution protocols to more than two distant users. Here we present a fully connected quantum network architecture in which a single entangled photon source distributes quantum states to many users while minimizing the resources required for each. Further, it does so without sacrificing security or functionality relative to two-party communication schemes. We demonstrate the feasibility of our approach using a single source of bipartite polarization entanglement, which is multiplexed into 12 wavelength channels. Six states are then distributed between four users in a fully connected graph using only one fibre and one polarization analysis module per user. Because no adaptations of the entanglement source are required to add users, the network can readily be scaled to a large number of users, without requiring trust in the provider of the source. Unlike previous attempts at multi-user networks, which have been based on active optical switches and therefore limited to some duty cycle, our implementation is fully passive and thus has the potential for unprecedented quantum communication speeds.