Stream-water temperature is a key variable controlling chemical, biological, and ecological processes in freshwater environments. Most models focus on a single river cross-section; however, temperature gradients along stretches and tributaries of a river network are crucial to assess ecohydrological features such as aquatic species suitability, growth and feeding rates, or disease transmission. We propose SESTET, a deterministic, spatially explicit stream temperature model for a whole river network, based on water and energy budgets at a reach scale and requiring only commonly available spatially distributed datasets, such as morphology and air temperature, as input. Heat exchange processes at the air-water interface are modelled via the widely used equilibrium temperature concept, whereas the effects of network structure are accounted for through advective heat fluxes. A case study was conducted on the prealpine Wigger river (Switzerland), where water temperatures have been measured in the period 2014-2018 at 11 spatially distributed locations. The results show the advantages of accounting for water and energy budgets at the reach scale for the entire river network, compared with simpler, lumped formulations. Because our approach fundamentally relies on spatially distributed air temperature fields, adequate spatial interpolation techniques that account for the effects of both elevation and thermal inversion in air temperature are key to a successful application of the model. SESTET allows the assessment of the magnitude of the various components of the heat budget at the reach scale and the derivation of reliable estimates of spatial gradients of mean daily stream temperatures for the whole catchment based on a limited number of conveniently located (viz., spanning the largest possible elevation range) measuring stations. Moreover, accounting for mixing processes and advective fluxes through the river network allows one to trust regionalized values of the parameters controlling the relationship between equilibrium and air temperature, a key feature to generalize the model to data-scarce catchments.