Stream temperature is a hydrological factor which affects the habitat suitability of many aquatic species, and is therefore of great concern in the actual context of climate change. Its prediction in ungauged basins is usually based on statistical approaches, such as multi-linear regression models or machine learning techniques. These models typically predict temperature metrics corresponding to yearly aggregates, such as the popular annual maximum weekly mean temperature (MWMT). As a consequence, they are often unable to predict the annual cycle of stream temperature, nor can the majority of them forecast the inter-annual variation of stream temperature. This study presents a new model to estimate the monthly mean stream temperature of ungauged rivers over multiple years in an Alpine country (Switzerland). Contrary to the current statistical approaches, this model rests upon the analytical solution of a simplified version of the energy-balance equation over an entire stream network. This physically-based approach presents some advantages, among which the possibility to interpret the model coefficients from a physical point of view, hereby enabling the restriction of their calibration range. The evaluation of the model over a new data set shows that the monthly mean stream temperature curve can be reproduced with a root mean square error of 1.3°C, which is similar in precision to the predictions obtained with standard multi-linear regression models. We illustrate through a simple example how the physical basis of the model can be used to gain more insight into the stream temperature dynamics at regional scales.